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Derivatives of inverse trig functions practice problems

arcsin, arccos and arctan, with the chain rule. 53 problems with worked solutions; in 52 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it; 20 of these also have a second solution, written separately and checked the same way. How we verify

Problem 2.410

hard Beautiful Video

Differentiate \( \displaystyle f(x) = \sqrt{1 - \left(x - 1\right)^{2}} + \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} \).

\[ \frac{d}{d x} \left(\sqrt{1 - \left(x - 1\right)^{2}} + \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} \]
sumApply the sum rule.✓ Proved
\[ = \left(x - 1\right) \frac{d}{d x} \operatorname{asin}{\left(x - 1 \right)} + \operatorname{asin}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} \]
productApply the product rule to the second term.✓ Proved
\[ = \left(x - 1\right) \frac{d}{d x} \operatorname{asin}{\left(x - 1 \right)} + \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} \]
derivative constant-multipleDifferentiate (x - 1). Simplify the constant factor.✓ Proved
\[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} + \frac{\left(x - 1\right) \frac{d}{d x} \left(x - 1\right)}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
inverse-trigApply the derivative rule for arcsine.✓ Proved
\[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
derivative algebra rewriteDifferentiate (x - 1). Simplify the fraction. Rewrite the square root as a power.✓ Proved
\[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} + \frac{\frac{d}{d x} \left(1 - \left(x - 1\right)^{2}\right)}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
chainApply the chain rule.✓ Proved
\[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} + \frac{\frac{d}{d x} 1 - \frac{d}{d x} \left(x - 1\right)^{2}}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
algebraDistribute the derivative over the subtraction.✓ Proved
\[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} - \frac{\frac{d}{d x} \left(x - 1\right)^{2}}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
derivativeDifferentiate the constant 1.✓ Proved
\[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{2 - 2 x}{2 \sqrt{1 - \left(x - 1\right)^{2}}} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
powerApply the power rule to (x - 1)**2.✓ Proved
\[ = \frac{1 - x}{\sqrt{1 - \left(x - 1\right)^{2}}} + \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
algebraSimplify the expression by multiplying terms.✓ Proved
\[ = \operatorname{asin}{\left(x - 1 \right)} \]
simplifyThe two terms involving the square root cancel each other out.✓ Proved

Answer \( \operatorname{asin}{\left(x - 1 \right)} \)

Top: the graph of f = sqrt(1 - (x - 1)²) + (x - 1)·asin(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (1 - x)/sqrt(1 - (x - 1)²) + asin(x - 1) + (x - 1)/sqrt(1 - (x - 1)²).

A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \left(\sqrt{1 - \left(x - 1\right)^{2}} + \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)}\right) \]
derivativeStart by writing the derivative of the given function with respect to x.Not checked
\[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} \]
sum rewrite rewriteApply the sum rule to differentiate each term separately. Rewrite the square root as a power of 1/2 for easier differentiation. Rewrite the power function using exp and log to handle the base and exponent structure.✓ Proved
\[ = \sqrt{1 - \left(x - 1\right)^{2}} \frac{d}{d x} \frac{\ln{\left(1 - \left(x - 1\right)^{2} \right)}}{2} + \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} \]
chain rewriteApply the chain rule to the exponential term. Rewrite exp(log(...)) back to the original radical form for clarity.✓ Proved
\[ = \frac{\sqrt{1 - \left(x - 1\right)^{2}} \frac{d}{d x} \ln{\left(1 - \left(x - 1\right)^{2} \right)}}{2} + \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} \]
constant-multiplePull out the constant factor 1/2 from the derivative.✓ Proved
\[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} + \frac{\frac{d}{d x} \left(1 - \left(x - 1\right)^{2}\right)}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
chainApply the chain rule to the logarithmic term.✓ Proved
\[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} + \frac{\frac{d}{d x} 1}{2 \sqrt{1 - \left(x - 1\right)^{2}}} + \frac{\frac{d}{d x} \left(- \left(x - 1\right)^{2}\right)}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
sumApply the sum rule to the inner function 1 - (x - 1)^2.✓ Proved
\[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} + \frac{\frac{d}{d x} \left(- \left(x - 1\right)^{2}\right)}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
constantThe derivative of the constant 1 is 0.✓ Proved
\[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} - \frac{\frac{d}{d x} \left(x - 1\right)^{2}}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
constant-multiplePull out the constant factor -1 from the derivative.✓ Proved
\[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} - \frac{2 x - 2}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
powerApply the power rule to (x-1)^2.✓ Proved
\[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} - \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
algebra algebraSimplify the coefficients in the first term. Simplify the fraction by canceling the factor of 2.✓ Proved
\[ = \frac{1 - x}{\sqrt{1 - \left(x - 1\right)^{2}}} + \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} \]
algebraSimplify the radical expression: sqrt(A)/A = 1/sqrt(A).✓ Proved
\[ = \frac{1 - x}{\sqrt{1 - \left(x - 1\right)^{2}}} + \left(x - 1\right) \frac{d}{d x} \operatorname{asin}{\left(x - 1 \right)} + \operatorname{asin}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) \]
productApply the product rule to the second term.✓ Proved
\[ = \frac{1 - x}{\sqrt{1 - \left(x - 1\right)^{2}}} + \left(x - 1\right) \frac{d}{d x} \operatorname{asin}{\left(x - 1 \right)} + \operatorname{asin}{\left(x - 1 \right)} \]
derivativeThe derivative of (x-1) with respect to x is 1.✓ Proved
\[ = \frac{1 - x}{\sqrt{1 - \left(x - 1\right)^{2}}} + \operatorname{asin}{\left(x - 1 \right)} + \frac{\left(x - 1\right) \frac{d}{d x} \left(x - 1\right)}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
inverse-trigApply the derivative formula for arcsine.✓ Proved
\[ = \frac{1 - x}{\sqrt{1 - \left(x - 1\right)^{2}}} + \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
derivative algebraThe derivative of (x-1) with respect to x is 1. Simplify the last term.✓ Proved
\[ = \operatorname{asin}{\left(x - 1 \right)} \]
simplifyThe terms involving the square root cancel each other out.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.1485

hard Beautiful Video

Differentiate \( \displaystyle f(x) = x \operatorname{atan}{\left(x \right)} - \frac{\ln{\left(x^{2} + 1 \right)}}{2} \).

\[ \frac{d}{d x} \left(x \operatorname{atan}{\left(x \right)} - \frac{\ln{\left(x^{2} + 1 \right)}}{2}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} x \operatorname{atan}{\left(x \right)} - \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]
sumApply the sum rule.✓ Proved
\[ = \frac{d}{d x} x \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
constantPull out the constant factor.✓ Proved
\[ = x \frac{d}{d x} \operatorname{atan}{\left(x \right)} + \operatorname{atan}{\left(x \right)} \frac{d}{d x} x - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} \operatorname{atan}{\left(x \right)} + \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
derivative simplifyDifferentiate x. Simplify the expression.✓ Proved
\[ = \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
inverse-trig algebraDifferentiate the arctangent function. Simplify the fraction.✓ Proved
\[ = \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \operatorname{atan}{\left(x \right)} \]
derivative algebra algebra simplifyDifferentiate the inner function x**2 + 1. Simplify the product. Combine the fractions. Simplify the final result.✓ Proved

Answer \( \operatorname{atan}{\left(x \right)} \)

Top: the graph of f = x·atan(x) - log(x² + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = atan(x).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \left(x \operatorname{atan}{\left(x \right)} - \frac{\ln{\left(x^{2} + 1 \right)}}{2}\right) \]
derivative✓ Proved
\[ = x \frac{d}{d x} \operatorname{atan}{\left(x \right)} + \operatorname{atan}{\left(x \right)} - \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]
product✓ Proved
\[ = \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]
derivative✓ Proved
\[ = \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
constant-multiple✓ Proved
\[ = \operatorname{atan}{\left(x \right)} \]
derivative simplify simplify✓ Proved

Problem 2.633

hard

Differentiate \( \displaystyle f(x) = \sqrt{1 - \left(4 x + 2\right)^{2}} + \left(4 x + 2\right) \operatorname{asin}{\left(4 x + 2 \right)} \).

\[ \frac{d}{d x} \left(\sqrt{1 - \left(4 x + 2\right)^{2}} + \left(4 x + 2\right) \operatorname{asin}{\left(4 x + 2 \right)}\right) \]
sumStart with the derivative of the entire function.✓ Proved
\[ = \frac{d}{d x} \left(4 x + 2\right) \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} \]
sumSplit the derivative into two parts.✓ Proved
\[ = \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{d}{d x} \left(4 x \operatorname{asin}{\left(4 x + 2 \right)} + 2 \operatorname{asin}{\left(4 x + 2 \right)}\right) \]
algebraExpand the second term.✓ Proved
\[ = \frac{d}{d x} 4 x \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{d}{d x} 2 \operatorname{asin}{\left(4 x + 2 \right)} \]
sumSplit the derivative of the expanded term.✓ Proved
\[ = 4 x \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + \operatorname{asin}{\left(4 x + 2 \right)} \frac{d}{d x} 4 x + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{d}{d x} 2 \operatorname{asin}{\left(4 x + 2 \right)} \]
productApply the product rule to the second term.✓ Proved
\[ = 4 x \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{d}{d x} 2 \operatorname{asin}{\left(4 x + 2 \right)} \]
derivativeDifferentiate 4*x.✓ Proved
\[ = 4 x \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + 2 \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} \]
constant-multipleDifferentiate 2*asin(4*x + 2) by pulling out the constant.✓ Proved
\[ = \left(4 x + 2\right) \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} \]
algebraCombine the terms involving the derivative of asin.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{\left(4 x + 2\right) \frac{d}{d x} \left(4 x + 2\right)}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
inverse-trigApply the derivative rule for arcsine.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{4 \left(4 x + 2\right)}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
derivativeDifferentiate 4*x + 2.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
algebra rewriteMultiply the terms in the last part. Rewrite the square root as a power.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{\frac{d}{d x} \left(1 - \left(4 x + 2\right)^{2}\right)}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} \]
chainApply the chain rule to the power term.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{\frac{d}{d x} 1 - \frac{d}{d x} \left(4 x + 2\right)^{2}}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} \]
sumSplit the derivative inside the power term.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} - \frac{\frac{d}{d x} \left(4 x + 2\right)^{2}}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} \]
derivative constant-multipleDifferentiate the constant 1. Handle the negative sign.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} - \frac{\left(8 x + 4\right) \frac{d}{d x} \left(4 x + 2\right)}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
powerApply the power rule to (4*x + 2)**2.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{- 32 x - 16}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
derivative algebraDifferentiate 4*x + 2. Multiply the terms.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{- 16 x - 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
algebraSimplify the first term.✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} \]
algebra algebra algebra simplifyCombine the fractions with the same denominator. Expand the numerator. Simplify the numerator. Final simplification.✓ Proved

Answer \( 4 \operatorname{asin}{\left(4 x + 2 \right)} \)

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \left(\sqrt{1 - \left(4 x + 2\right)^{2}} + \left(4 x + 2\right) \operatorname{asin}{\left(4 x + 2 \right)}\right) \]
derivative✓ Proved
\[ = \frac{d}{d x} \left(4 x + 2\right) \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} \]
sum✓ Proved
\[ = \left(4 x + 2\right) \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + \operatorname{asin}{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right) + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} \]
product✓ Proved
\[ = \left(4 x + 2\right) \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} \]
constant-multiple✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{\left(4 x + 2\right) \frac{d}{d x} \left(4 x + 2\right)}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
chain✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{4 \left(4 x + 2\right)}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
constant-multiple✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
simplify✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{\frac{d}{d x} \left(1 - \left(4 x + 2\right)^{2}\right)}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} \]
chain✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{\left(- 8 x - 4\right) \frac{d}{d x} \left(4 x + 2\right)}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
product✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{- 32 x - 16}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
constant-multiple simplify✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{- 16 x - 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
simplify✓ Proved
\[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} \]
simplifyThe two terms with sqrt cancel out.✓ Proved

Problem 2.816

hard

Differentiate \( \displaystyle f(x) = \frac{\operatorname{acos}{\left(2 x \right)}}{2 x} \).

\[ \frac{d}{d x} \frac{\operatorname{acos}{\left(2 x \right)}}{2 x} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \operatorname{acos}{\left(2 x \right)} \frac{d}{d x} \frac{1}{2 x} + \frac{\frac{d}{d x} \operatorname{acos}{\left(2 x \right)}}{2 x} \]
quotientApply the quotient rule.✓ Proved
\[ = \operatorname{acos}{\left(2 x \right)} \frac{d}{d x} \frac{1}{2 x} - \frac{\frac{d}{d x} 2 x}{2 x \sqrt{1 - 4 x^{2}}} \]
chainApply the chain rule to the numerator.✓ Proved
\[ = \operatorname{acos}{\left(2 x \right)} \frac{d}{d x} \frac{1}{2 x} - \frac{1}{x \sqrt{1 - 4 x^{2}}} \]
derivative algebra rewriteDifferentiate the inner function 2*x. Simplify the fraction. Rewrite 1/(2*x) as a power.✓ Proved
\[ = - \frac{1}{x \sqrt{1 - 4 x^{2}}} - \frac{\operatorname{acos}{\left(2 x \right)} \frac{d}{d x} x}{2 x^{2}} \]
powerApply the power rule.✓ Proved
\[ = - \frac{1}{x \sqrt{1 - 4 x^{2}}} - \frac{\operatorname{acos}{\left(2 x \right)}}{2 x^{2}} \]
derivative algebra simplifyDifferentiate x. Simplify the expression. Final simplification.✓ Proved

Answer \( - \frac{1}{x \sqrt{1 - 4 x^{2}}} - \frac{\operatorname{acos}{\left(2 x \right)}}{2 x^{2}} \)

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \frac{\operatorname{acos}{\left(2 x \right)}}{2 x} \]
derivative✓ Proved
\[ = \frac{2 x \frac{d}{d x} \operatorname{acos}{\left(2 x \right)} - \operatorname{acos}{\left(2 x \right)} \frac{d}{d x} 2 x}{4 x^{2}} \]
quotientApply the quotient rule✓ Proved
\[ = \frac{- \frac{4 x}{\sqrt{1 - 4 x^{2}}} - 2 \operatorname{acos}{\left(2 x \right)}}{4 x^{2}} \]
chain simplifyDifferentiate acos(2*x) using chain rule Simplify the derivative of acos(2*x)✓ Proved
\[ = - \frac{1}{x \sqrt{1 - 4 x^{2}}} - \frac{\operatorname{acos}{\left(2 x \right)}}{2 x^{2}} \]
simplify simplify simplifySeparate the terms Simplify (2*x)**2 to 4*x**2 Cancel factors✓ Proved

Problem 2.326

hard Video

Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\left(5 x - 3\right)^{2} \right)} \).

\[ \frac{d}{d x} \operatorname{atan}{\left(\left(5 x - 3\right)^{2} \right)} \]
rewriteStart with the derivative of the function. Rewrite the power using the exponential and logarithm identity.✓ Proved
\[ = \frac{\frac{d}{d x} \left(5 x - 3\right)^{2}}{\left(5 x - 3\right)^{4} + 1} \]
chain algebraApply the chain rule for the arctangent function. Simplify the denominator by converting the exponential back to a power.✓ Proved
\[ = \frac{2 \left(5 x - 3\right)^{2} \frac{d}{d x} \ln{\left(5 x - 3 \right)}}{\left(5 x - 3\right)^{4} + 1} \]
chainApply the chain rule to the numerator.✓ Proved
\[ = \frac{50 x - 30}{\left(5 x - 3\right)^{4} + 1} \]
chain constant-multiple algebra simplifyDifferentiate the logarithm term using the chain rule. Simplify the constants in the numerator. Convert the exponential back to a power for simplification. Simplify the fraction by canceling the common term (5*x - 3).✓ Proved

Answer \( \frac{50 x - 30}{\left(5 x - 3\right)^{4} + 1} \)

Top: the graph of f = atan((5x - 3)²). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (50x - 30)/((5x - 3)⁴ + 1).

A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \operatorname{atan}{\left(\left(5 x - 3\right)^{2} \right)} \]
derivativeStart with the derivative of the function.Not checked
\[ = \frac{\frac{d}{d x} \left(5 x - 3\right)^{2}}{\left(5 x - 3\right)^{4} + 1} \]
chain algebraApply the chain rule for atan(u): derivative is u'/(u^2+1). Simplify the denominator ((5*x-3)^2)^2 to (5*x-3)^4.✓ Proved
\[ = \frac{\left(10 x - 6\right) \frac{d}{d x} \left(5 x - 3\right)}{\left(5 x - 3\right)^{4} + 1} \]
chainApply the chain rule to the inner square function.✓ Proved
\[ = \frac{5 \left(10 x - 6\right)}{\left(5 x - 3\right)^{4} + 1} \]
constant-multipleDifferentiate the linear term 5*x - 3.✓ Proved
\[ = \frac{50 x - 30}{\left(5 x - 3\right)^{4} + 1} \]
simplifyCombine the constants and numerator terms.✓ Proved

Problem 2.817

hard Video

Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\frac{1}{x} \right)} \).

\[ \frac{d}{d x} \operatorname{atan}{\left(\frac{1}{x} \right)} \]
derivative algebraStart with the derivative of the function. Rewrite 1/x as x**-1.✓ Proved
\[ = \frac{\frac{d}{d x} \frac{1}{x}}{1 + \frac{1}{x^{2}}} \]
inverse-trig powerApply the derivative rule for arctan(u). Simplify the squared term.✓ Proved
\[ = - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)} \]
power algebra algebraDifferentiate x**-1 using the power rule. Combine the fraction and the derivative. Rewrite x**-2 as 1/x**2.✓ Proved
\[ = - \frac{1}{x^{2} + 1} \]
algebra simplifySimplify the complex fraction by multiplying by the reciprocal. Cancel the x**2 terms to get the final result.✓ Proved

Answer \( - \frac{1}{x^{2} + 1} \)

Top: the graph of f = atan(1/x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -1/(x²·(1 + x^(-2))).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \operatorname{atan}{\left(\frac{1}{x} \right)} \]
derivative✓ Proved
\[ = \frac{\frac{d}{d x} \frac{1}{x}}{1 + \frac{1}{x^{2}}} \]
chainApply chain rule to atan✓ Proved
\[ = - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)} \]
derivative simplifyDifferentiate 1/x as x**-1 Remove multiplication sign✓ Proved
\[ = - \frac{1}{x^{2} + 1} \]
simplifySimplify fraction✓ Proved

Problem 2.385

hard Video

Differentiate \( \displaystyle f(x) = \frac{\operatorname{acos}{\left(x \right)}}{x} \).

\[ \frac{d}{d x} \frac{\operatorname{acos}{\left(x \right)}}{x} \]
derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product.✓ Proved
\[ = \operatorname{acos}{\left(x \right)} \frac{d}{d x} \frac{1}{x} + \frac{\frac{d}{d x} \operatorname{acos}{\left(x \right)}}{x} \]
productApply the product rule.✓ Proved
\[ = \operatorname{acos}{\left(x \right)} \frac{d}{d x} \frac{1}{x} - \frac{1}{x \sqrt{1 - x^{2}}} \]
derivativeDifferentiate the first term.✓ Proved
\[ = - \frac{1}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2}} \]
derivative algebraDifferentiate the second term. Simplify the expression.✓ Proved
\[ = \frac{- x - \sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}{x^{2} \sqrt{1 - x^{2}}} \]
algebraCombine into a single fraction.✓ Proved

Answer \( - \frac{1}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2}} \)

Top: the graph of f = acos(x)/x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -1/(x·sqrt(1 - x²)) - acos(x)/x².

A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \frac{\operatorname{acos}{\left(x \right)}}{x} \]
derivative rewriteStart by identifying the function to differentiate. Rewrite the division as multiplication by a negative power to prepare for the product rule.✓ Proved
\[ = \operatorname{acos}{\left(x \right)} \frac{d}{d x} \frac{1}{x} + \frac{\frac{d}{d x} \operatorname{acos}{\left(x \right)}}{x} \]
productApply the product rule: (uv)' = u'v + uv'.✓ Proved
\[ = \operatorname{acos}{\left(x \right)} \frac{d}{d x} \frac{1}{x} - \frac{1}{x \sqrt{1 - x^{2}}} \]
trigDifferentiate the inverse cosine term: d/dx(acos(x)) = -1/sqrt(1-x^2).✓ Proved
\[ = - \frac{1}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2}} \]
power simplifyDifferentiate the power term: d/dx(x^n) = n*x^(n-1). Simplify the expressions by combining terms and removing negative exponents.✓ Proved

Problem 2.406

hard Video

Differentiate \( \displaystyle f(x) = \ln{\left(\operatorname{atan}{\left(4 x - 1 \right)} \right)} \).

\[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(4 x - 1 \right)} \right)} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(4 x - 1 \right)}}{\operatorname{atan}{\left(4 x - 1 \right)}} \]
logarithmicApply the chain rule for the natural logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \left(4 x - 1\right)}{\left(\left(4 x - 1\right)^{2} + 1\right) \operatorname{atan}{\left(4 x - 1 \right)}} \]
inverse-trigApply the chain rule for the arctangent function.✓ Proved
\[ = \frac{4}{\left(\left(4 x - 1\right)^{2} + 1\right) \operatorname{atan}{\left(4 x - 1 \right)}} \]
derivative algebraDifferentiate the linear term inside the parentheses. Simplify the expression by multiplying the terms.✓ Proved
\[ = \frac{4}{\left(16 x^{2} - 8 x + 2\right) \operatorname{atan}{\left(4 x - 1 \right)}} \]
algebra simplifyExpand the squared binomial. Combine like terms in the denominator.✓ Proved

Answer \( \frac{4}{\left(\left(4 x - 1\right)^{2} + 1\right) \operatorname{atan}{\left(4 x - 1 \right)}} \)

Top: the graph of f = log(atan(4x - 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4/(((4x - 1)² + 1)·atan(4x - 1)).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(4 x - 1 \right)} \right)} \]
derivative✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(4 x - 1 \right)}}{\operatorname{atan}{\left(4 x - 1 \right)}} \]
chainDerivative of log(u) is u' / u✓ Proved
\[ = \frac{4}{\left(\left(4 x - 1\right)^{2} + 1\right) \operatorname{atan}{\left(4 x - 1 \right)}} \]
derivative simplifyDerivative of atan(v) is v'/(1+v**2), v=4*x-1 Combine fractions✓ Proved

Problem 2.553

hard Video

Differentiate \( \displaystyle f(x) = e^{\operatorname{atan}{\left(3 x + 1 \right)}} \).

\[ \frac{d}{d x} e^{\operatorname{atan}{\left(3 x + 1 \right)}} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = e^{\operatorname{atan}{\left(3 x + 1 \right)}} \frac{d}{d x} \operatorname{atan}{\left(3 x + 1 \right)} \]
chainApply the chain rule to the exponential function.✓ Proved
\[ = \frac{e^{\operatorname{atan}{\left(3 x + 1 \right)}} \frac{d}{d x} \left(3 x + 1\right)}{\left(3 x + 1\right)^{2} + 1} \]
trigDifferentiate the arctangent function.✓ Proved
\[ = \frac{3 e^{\operatorname{atan}{\left(3 x + 1 \right)}}}{\left(3 x + 1\right)^{2} + 1} \]
derivative algebraDifferentiate the linear inner function. Rearrange the expression.✓ Proved
\[ = \frac{3 e^{\operatorname{atan}{\left(3 x + 1 \right)}}}{9 x^{2} + 6 x + 2} \]
algebra simplifyExpand the denominator. Combine the constant terms in the denominator.✓ Proved

Answer \( \frac{3 e^{\operatorname{atan}{\left(3 x + 1 \right)}}}{\left(3 x + 1\right)^{2} + 1} \)

Top: the graph of f = e^(atan(3x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3e^(atan(3x + 1))/((3x + 1)² + 1).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} e^{\operatorname{atan}{\left(3 x + 1 \right)}} \]
derivative✓ Proved
\[ = e^{\operatorname{atan}{\left(3 x + 1 \right)}} \frac{d}{d x} \operatorname{atan}{\left(3 x + 1 \right)} \]
chainApply chain rule to exp✓ Proved
\[ = \frac{3 e^{\operatorname{atan}{\left(3 x + 1 \right)}}}{\left(3 x + 1\right)^{2} + 1} \]
derivative simplifyDifferentiate atan(3*x+1) Simplify the fraction✓ Proved

Problem 2.838

hard Video

Differentiate \( \displaystyle f(x) = \ln{\left(\operatorname{atan}{\left(2 x + 1 \right)} \right)} \).

\[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(2 x + 1 \right)} \right)} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(2 x + 1 \right)}}{\operatorname{atan}{\left(2 x + 1 \right)}} \]
logarithmicApply the chain rule for the natural logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \left(2 x + 1\right)}{\left(\left(2 x + 1\right)^{2} + 1\right) \operatorname{atan}{\left(2 x + 1 \right)}} \]
chainApply the chain rule for the arctangent function.✓ Proved
\[ = \frac{2}{\left(\left(2 x + 1\right)^{2} + 1\right) \operatorname{atan}{\left(2 x + 1 \right)}} \]
derivative algebraDifferentiate the inner linear function. Combine the terms into a single fraction.✓ Proved
\[ = \frac{2}{\left(4 x^{2} + 4 x + 2\right) \operatorname{atan}{\left(2 x + 1 \right)}} \]
algebra simplifyExpand the squared binomial. Simplify the expression in the denominator.✓ Proved

Answer \( \frac{2}{\left(\left(2 x + 1\right)^{2} + 1\right) \operatorname{atan}{\left(2 x + 1 \right)}} \)

Top: the graph of f = log(atan(2x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2/(((2x + 1)² + 1)·atan(2x + 1)).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(2 x + 1 \right)} \right)} \]
derivativeStart with the derivative of the whole function✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(2 x + 1 \right)}}{\operatorname{atan}{\left(2 x + 1 \right)}} \]
chainApply the chain rule to log✓ Proved
\[ = \frac{2}{\left(\left(2 x + 1\right)^{2} + 1\right) \operatorname{atan}{\left(2 x + 1 \right)}} \]
chain simplifyDifferentiate atan(2*x + 1) Combine the fractions✓ Proved
\[ = \frac{2}{\left(4 x^{2} + 4 x + 2\right) \operatorname{atan}{\left(2 x + 1 \right)}} \]
algebraExpand the square in the denominator✓ Proved
\[ = \frac{1}{\left(2 x^{2} + 2 x + 1\right) \operatorname{atan}{\left(2 x + 1 \right)}} \]
simplifyFactor out 2 and cancel✓ Proved

Problem 2.377

hard Video

Differentiate \( \displaystyle f(x) = e^{\operatorname{atan}{\left(x - 3 \right)}} \).

\[ \frac{d}{d x} e^{\operatorname{atan}{\left(x - 3 \right)}} \]
Start with the derivative of the function.✓ Proved
\[ = e^{\operatorname{atan}{\left(x - 3 \right)}} \frac{d}{d x} \operatorname{atan}{\left(x - 3 \right)} \]
chainApply the chain rule.✓ Proved
\[ = \frac{e^{\operatorname{atan}{\left(x - 3 \right)}}}{\left(x - 3\right)^{2} + 1} \]
derivative algebraDifferentiate the inner function atan(x - 3). Rewrite the expression as a single fraction.✓ Proved
\[ = \frac{e^{\operatorname{atan}{\left(x - 3 \right)}}}{x^{2} - 6 x + 10} \]
algebra simplifyExpand the denominator. Simplify the denominator.✓ Proved

Answer \( \frac{e^{\operatorname{atan}{\left(x - 3 \right)}}}{\left(x - 3\right)^{2} + 1} \)

Top: the graph of f = e^(atan(x - 3)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = e^(atan(x - 3))/((x - 3)² + 1).

A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} e^{\operatorname{atan}{\left(x - 3 \right)}} \]
derivativeStart by writing the derivative of the given function with respect to x.Not checked
\[ = e^{\operatorname{atan}{\left(x - 3 \right)}} \frac{d}{d x} \operatorname{atan}{\left(x - 3 \right)} \]
chainApply the chain rule: the derivative of exp(u) is exp(u) times the derivative of u.✓ Proved
\[ = \frac{e^{\operatorname{atan}{\left(x - 3 \right)}} \frac{d}{d x} \left(x - 3\right)}{\left(x - 3\right)^{2} + 1} \]
chainApply the chain rule again for atan(v): the derivative is 1/(v^2+1) times the derivative of v.✓ Proved
\[ = \frac{e^{\operatorname{atan}{\left(x - 3 \right)}}}{\left(x - 3\right)^{2} + 1} \]
derivative simplifyThe derivative of x - 3 with respect to x is 1. Simplify the expression by removing the multiplication by 1.✓ Proved

Problem 2.481

hard

Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(\sqrt{x} \right)} \).

\[ \frac{d}{d x} \operatorname{asin}{\left(\sqrt{x} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sqrt{x}}{\sqrt{1 - x}} \]
chain algebraApply the chain rule to the arcsine function. Simplify the term inside the square root.✓ Proved
\[ = \frac{1}{2 \sqrt{x} \sqrt{1 - x}} \]
derivative algebraDifferentiate the inner square root function. Combine the terms into a single fraction.✓ Proved
\[ = \frac{1}{2 \sqrt{x \left(1 - x\right)}} \]
simplifySimplify the expression by bringing x into the square root.≈ Checked numerically

Answer \( \frac{1}{2 \sqrt{x} \sqrt{1 - x}} \)

A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \operatorname{asin}{\left(\sqrt{x} \right)} \]
derivativeStart with the derivative of the composite function.Not checked
\[ = \frac{\frac{d}{d x} \sqrt{x}}{\sqrt{1 - x}} \]
chain algebraApply the chain rule for the outer arcsine function. Simplify the term inside the square root: (sqrt(x))**2 = x.✓ Proved
\[ = \frac{1}{2 \sqrt{x} \sqrt{1 - x}} \]
power algebraDifferentiate the inner square root function using the power rule. Combine the constants and terms into a single fraction.✓ Proved

✓ Nihil obstat Lines: 5 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.353

hard Video

Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\sin{\left(2 x \right)} \right)} \).

\[ \frac{d}{d x} \operatorname{atan}{\left(\sin{\left(2 x \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} + 1} \]
chainApply the chain rule for the arctangent function.✓ Proved
\[ = \frac{\cos{\left(2 x \right)} \frac{d}{d x} 2 x}{\sin^{2}{\left(2 x \right)} + 1} \]
chainApply the chain rule for the sine function.✓ Proved
\[ = \frac{2 \cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} + 1} \]
derivative algebraDifferentiate the inner linear function. Simplify the expression by combining terms.✓ Proved

Answer \( \frac{2 \cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} + 1} \)

Top: the graph of f = atan(sin(2x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2cos(2x)/(sin(2x)² + 1).

A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \operatorname{atan}{\left(\sin{\left(2 x \right)} \right)} \]
derivativeStart by identifying the function to be differentiated.Not checked
\[ = \frac{\frac{d}{d x} \sin{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} + 1} \]
chainApply the chain rule for the outer arctangent function: d/dx atan(u) = u' / (u^2 + 1).✓ Proved
\[ = \frac{\cos{\left(2 x \right)} \frac{d}{d x} 2 x}{\sin^{2}{\left(2 x \right)} + 1} \]
chainApply the chain rule for the inner sine function: d/dx sin(v) = v' cos(v).✓ Proved
\[ = \frac{2 \cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} + 1} \]
constant-multiple simplifyDifferentiate the innermost term 2*x with respect to x. Rearrange the terms into a standard fractional form.✓ Proved

Problem 2.400

hard Video

Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\sin{\left(x - 1 \right)} \right)} \).

\[ \frac{d}{d x} \operatorname{atan}{\left(\sin{\left(x - 1 \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(x - 1 \right)}}{\sin^{2}{\left(x - 1 \right)} + 1} \]
chainApply the chain rule for the arctangent function.✓ Proved
\[ = \frac{\cos{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)}{\sin^{2}{\left(x - 1 \right)} + 1} \]
trigApply the chain rule for the sine function.✓ Proved
\[ = \frac{\cos{\left(x - 1 \right)}}{\sin^{2}{\left(x - 1 \right)} + 1} \]
derivative simplifyDifferentiate the inner linear function. Simplify the expression.✓ Proved

Answer \( \frac{\cos{\left(x - 1 \right)}}{\sin^{2}{\left(x - 1 \right)} + 1} \)

Top: the graph of f = atan(sin(x - 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = cos(x - 1)/(sin(x - 1)² + 1).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \operatorname{atan}{\left(\sin{\left(x - 1 \right)} \right)} \]
derivative✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(x - 1 \right)}}{\sin^{2}{\left(x - 1 \right)} + 1} \]
chainApply derivative of atan(u) = u'/(1+u^2)✓ Proved
\[ = \frac{\cos{\left(x - 1 \right)}}{\sin^{2}{\left(x - 1 \right)} + 1} \]
derivativeDerivative of sin(x-1) is cos(x-1)✓ Proved

Problem 2.465

hard Video

Differentiate \( \displaystyle f(x) = \ln{\left(\operatorname{atan}{\left(2 x \right)} \right)} \).

\[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(2 x \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(2 x \right)}}{\operatorname{atan}{\left(2 x \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} 2 x}{\left(4 x^{2} + 1\right) \operatorname{atan}{\left(2 x \right)}} \]
chainApply the chain rule for the arctangent.✓ Proved
\[ = \frac{2}{\left(4 x^{2} + 1\right) \operatorname{atan}{\left(2 x \right)}} \]
derivative algebraDifferentiate the innermost function 2*x. Simplify the expression by multiplying the terms.✓ Proved

Answer \( \frac{2}{\left(4 x^{2} + 1\right) \operatorname{atan}{\left(2 x \right)}} \)

Top: the graph of f = log(atan(2x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2/((4x² + 1)·atan(2x)).

A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(2 x \right)} \right)} \]
derivativeStart by identifying the function to differentiate.Not checked
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(2 x \right)}}{\operatorname{atan}{\left(2 x \right)}} \]
chainApply the chain rule for the outer logarithm function.✓ Proved
\[ = \frac{\frac{d}{d x} 2 x}{\left(4 x^{2} + 1\right) \operatorname{atan}{\left(2 x \right)}} \]
chainApply the chain rule for the inner arctangent function.✓ Proved
\[ = \frac{2}{\left(4 x^{2} + 1\right) \operatorname{atan}{\left(2 x \right)}} \]
constant-multiple algebraDifferentiate the linear term 2*x with respect to x. Combine the terms into a single fraction.✓ Proved

Problem 2.497

hard Video

Differentiate \( \displaystyle f(x) = \ln{\left(\operatorname{atan}{\left(4 x + 2 \right)} \right)} \).

\[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(4 x + 2 \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(4 x + 2 \right)}}{\operatorname{atan}{\left(4 x + 2 \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \left(4 x + 2\right)}{\left(\left(4 x + 2\right)^{2} + 1\right) \operatorname{atan}{\left(4 x + 2 \right)}} \]
chainApply the chain rule for the arctangent.✓ Proved
\[ = \frac{4}{\left(\left(4 x + 2\right)^{2} + 1\right) \operatorname{atan}{\left(4 x + 2 \right)}} \]
derivative algebraDifferentiate the linear inner function. Combine the terms into a single fraction.✓ Proved

Answer \( \frac{4}{\left(\left(4 x + 2\right)^{2} + 1\right) \operatorname{atan}{\left(4 x + 2 \right)}} \)

Top: the graph of f = log(atan(4x + 2)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4/(((4x + 2)² + 1)·atan(4x + 2)).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(4 x + 2 \right)} \right)} \]
derivative✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(4 x + 2 \right)}}{\operatorname{atan}{\left(4 x + 2 \right)}} \]
chainApply chain rule to log✓ Proved
\[ = \frac{4}{\left(\left(4 x + 2\right)^{2} + 1\right) \operatorname{atan}{\left(4 x + 2 \right)}} \]
derivative simplifyDifferentiate atan(4*x + 2) Combine fractions✓ Proved

Problem 2.820

hard Video

Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\sin{\left(5 x + 1 \right)} \right)} \).

\[ \frac{d}{d x} \operatorname{atan}{\left(\sin{\left(5 x + 1 \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(5 x + 1 \right)}}{\sin^{2}{\left(5 x + 1 \right)} + 1} \]
chainApply the chain rule for the arctangent function.✓ Proved
\[ = \frac{\cos{\left(5 x + 1 \right)} \frac{d}{d x} \left(5 x + 1\right)}{\sin^{2}{\left(5 x + 1 \right)} + 1} \]
chainApply the chain rule for the sine function.✓ Proved
\[ = \frac{5 \cos{\left(5 x + 1 \right)}}{\sin^{2}{\left(5 x + 1 \right)} + 1} \]
derivative algebraDifferentiate the inner linear function. Simplify the expression by combining terms.✓ Proved

Answer \( \frac{5 \cos{\left(5 x + 1 \right)}}{\sin^{2}{\left(5 x + 1 \right)} + 1} \)

Top: the graph of f = atan(sin(5x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5cos(5x + 1)/(sin(5x + 1)² + 1).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \operatorname{atan}{\left(\sin{\left(5 x + 1 \right)} \right)} \]
derivative✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(5 x + 1 \right)}}{\sin^{2}{\left(5 x + 1 \right)} + 1} \]
chainDerivative of atan(u) is u'/(1+u^2)✓ Proved
\[ = \frac{\cos{\left(5 x + 1 \right)} \frac{d}{d x} \left(5 x + 1\right)}{\sin^{2}{\left(5 x + 1 \right)} + 1} \]
chainDerivative of sin(v) is cos(v)*v'✓ Proved
\[ = \frac{5 \cos{\left(5 x + 1 \right)}}{\sin^{2}{\left(5 x + 1 \right)} + 1} \]
constant-multiple simplifyDerivative of 5*x + 1 is 5 Final simplified derivative✓ Proved

Problem 2.548

hard Video

Differentiate \( \displaystyle f(x) = e^{\operatorname{atan}{\left(x \right)}} \).

\[ \frac{d}{d x} e^{\operatorname{atan}{\left(x \right)}} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = e^{\operatorname{atan}{\left(x \right)}} \frac{d}{d x} \operatorname{atan}{\left(x \right)} \]
chainApply the chain rule.✓ Proved
\[ = \frac{e^{\operatorname{atan}{\left(x \right)}}}{x^{2} + 1} \]
derivative algebraDifferentiate the inner function atan(x). Rearrange the terms for clarity.✓ Proved

Answer \( \frac{e^{\operatorname{atan}{\left(x \right)}}}{x^{2} + 1} \)

Top: the graph of f = e^(atan(x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = e^(atan(x))/(x² + 1).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} e^{\operatorname{atan}{\left(x \right)}} \]
derivative✓ Proved
\[ = e^{\operatorname{atan}{\left(x \right)}} \frac{d}{d x} \operatorname{atan}{\left(x \right)} \]
chainApply chain rule to exp(atan(x))✓ Proved
\[ = \frac{e^{\operatorname{atan}{\left(x \right)}}}{x^{2} + 1} \]
derivativeDerivative of atan(x) is 1/(x**2+1)✓ Proved

Problem 2.607

hard Video

Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(4 x^{2} \right)} \).

\[ \frac{d}{d x} \operatorname{atan}{\left(4 x^{2} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} 4 x^{2}}{16 x^{4} + 1} \]
chainApply the chain rule for the arctangent function.✓ Proved
\[ = \frac{8 x}{16 x^{4} + 1} \]
product algebraDifferentiate the inner function 4*x**2. Simplify the denominator by squaring the term.✓ Proved

Answer \( \frac{8 x}{16 x^{4} + 1} \)

Top: the graph of f = atan(4x²). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 8x/(16x⁴ + 1).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \operatorname{atan}{\left(4 x^{2} \right)} \]
derivative✓ Proved
\[ = \frac{8 x}{16 x^{4} + 1} \]
chain simplifyApply chain rule to atan(4*x**2) Simplify (4*x**2)**2 to 16*x**4✓ Proved

Problem 2.571

hard Video

Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\left(x - 1\right)^{2} \right)} \).

\[ \frac{d}{d x} \operatorname{atan}{\left(\left(x - 1\right)^{2} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \left(x - 1\right)^{2}}{\left(x - 1\right)^{4} + 1} \]
chainApply the chain rule for the arctangent function.✓ Proved
\[ = \frac{2 x - 2}{\left(x - 1\right)^{4} + 1} \]
powerDifferentiate the inner function (x - 1)**2 using the power rule.✓ Proved

Answer \( \frac{2 x - 2}{\left(x - 1\right)^{4} + 1} \)

Top: the graph of f = atan((x - 1)²). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (2x - 2)/((x - 1)⁴ + 1).

A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

\[ \frac{d}{d x} \operatorname{atan}{\left(\left(x - 1\right)^{2} \right)} \]
derivative✓ Proved
\[ = \frac{2 x - 2}{\left(x - 1\right)^{4} + 1} \]
chain powerApply chain rule to atan(u) with u = (x-1)**2 Simplify ((x-1)**2)**2 to (x-1)**4✓ Proved