Derivatives of exponential functions practice problems
eˣ and its relatives, usually with the chain rule close behind. 65 problems with worked solutions; in 65 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
algebraMultiply the numerator and denominator by 2 * (x - 3)**(1/2) is not quite right, let's use the previous step's logic: rewrite (x-3)**(1/2) as (x-3)/(x-3)**(1/2).✓ Proved
Top: the graph of f = sqrt(x - 3)·e^(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sqrt(x - 3)·e^(x - 3) + e^(x - 3)/(2sqrt(x - 3)).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \sqrt{x - 3} e^{x - 3} \]
derivativeStart by identifying the function to be differentiated.Not checked
derivativesimplifyalgebraDifferentiate the inner linear functions x - 3. Simplify the powers and fractions. Rewrite the first term to have a common denominator structure.✓ Proved
algebrasimplifyCombine the terms over a common denominator. Expand and simplify the numerator.✓ 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
Differentiate \( \displaystyle f(x) = \frac{e^{3 x + 1}}{3 x + 2} \).
\[ \frac{d}{d x} \frac{e^{3 x + 1}}{3 x + 2} \]
derivativerewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
\[ = e^{3 x + 1} \frac{d}{d x} \frac{1}{3 x + 2} + \frac{\frac{d}{d x} e^{3 x + 1}}{3 x + 2} \]
productApply the product rule.✓ Proved
\[ = e^{3 x + 1} \frac{d}{d x} \frac{1}{3 x + 2} + \frac{3 e^{3 x + 1}}{3 x + 2} \]
chainDifferentiate the first part using the chain rule.✓ Proved
\[ = \frac{3 e^{3 x + 1}}{3 x + 2} - \frac{e^{3 x + 1} \frac{d}{d x} \left(3 x + 2\right)}{\left(3 x + 2\right)^{2}} \]
chainDifferentiate the second part using the chain rule.✓ Proved
\[ = \frac{3 e^{3 x + 1}}{3 x + 2} - \frac{3 e^{3 x + 1}}{\left(3 x + 2\right)^{2}} \]
derivativealgebraalgebraEvaluate the derivative of the inner function. Simplify the expression. Rewrite with positive exponents.✓ Proved
\[ = \frac{3 \left(3 x + 2\right) e^{3 x + 1} - 3 e^{3 x + 1}}{\left(3 x + 2\right)^{2}} \]
algebraFind a common denominator.✓ Proved
\[ = \frac{3 \left(3 x + 1\right) e^{3 x + 1}}{\left(3 x + 2\right)^{2}} \]
algebrasimplifyFactor out the common term 3 * exp(3*x + 1). Simplify the expression inside the parentheses.✓ Proved
Answer \( \frac{\left(9 x + 3\right) e^{3 x + 1}}{9 x^{2} + 12 x + 4} \)
Top: the graph of f = e^(3x + 1)/(3x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3e^(3x + 1)/(3x + 2) - 3e^(3x + 1)/(3x + 2)².
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{e^{3 x + 1}}{3 x + 2} \]
derivativeStart by identifying the function to differentiate.Not checked
\[ = \frac{\left(3 x + 2\right) \frac{d}{d x} e^{3 x + 1} - e^{3 x + 1} \frac{d}{d x} \left(3 x + 2\right)}{\left(3 x + 2\right)^{2}} \]
quotientApply the quotient rule: (u'v - uv') / v^2.✓ Proved
\[ = \frac{\left(3 x + 2\right) e^{3 x + 1} \frac{d}{d x} \left(3 x + 1\right) - e^{3 x + 1} \frac{d}{d x} \left(3 x + 2\right)}{\left(3 x + 2\right)^{2}} \]
chainDifferentiate the numerator using the chain rule for the exponential function.✓ Proved
\[ = \frac{3 \left(3 x + 2\right) e^{3 x + 1} - e^{3 x + 1} \frac{d}{d x} \left(3 x + 2\right)}{\left(3 x + 2\right)^{2}} \]
chainDifferentiate the inner function 3*x + 1.✓ Proved
\[ = \frac{3 \left(3 x + 2\right) e^{3 x + 1} - 3 e^{3 x + 1}}{\left(3 x + 2\right)^{2}} \]
chainalgebraDifferentiate the inner function 3*x + 2. Factor out common terms in the numerator.✓ Proved
\[ = \frac{3 \left(3 x + 1\right) e^{3 x + 1}}{\left(3 x + 2\right)^{2}} \]
algebrasimplifySimplify the expression inside the parentheses. Combine constants and simplify the final result.✓ 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
Top: the graph of f = (2x - 1)²·e^(1 - 2x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2(2x - 1)²·e^(1 - 2x) + (8x - 4)·e^(1 - 2x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
rewriterewritealgebraStart with the derivative of the function. Rewrite the square root as a power. Rewrite the power using the exponential and logarithm. Combine the exponents using properties of exponents.✓ Proved
Top: the graph of f = sqrt(x)·e^(x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sqrt(x)·e^(x) + e^(x)/(2sqrt(x)).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \sqrt{x} e^{x} \]
derivativerewriteStart by identifying the function to differentiate. Rewrite sqrt(x) as x**(1/2) for easier handling with power rules.✓ Proved
powerDifferentiate the first term x**(1/2) using the power rule.✓ Proved
\[ = \sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}} \]
exponentialrewriteDifferentiate the second term exp(x), whose derivative is itself. Rewrite negative exponents and fractional powers back to radical form for clarity.✓ Proved
\[ = \frac{\left(2 x + 1\right) e^{x}}{2 \sqrt{x}} \]
simplifyCombine the terms inside the parenthesis over a common denominator.✓ 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
rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
\[ = e^{5 x} \frac{d}{d x} \frac{1}{5 x + 1} + \frac{\frac{d}{d x} e^{5 x}}{5 x + 1} \]
productApply the product rule.✓ Proved
\[ = e^{5 x} \frac{d}{d x} \frac{1}{5 x + 1} + \frac{5 e^{5 x}}{5 x + 1} \]
derivativeDifferentiate the first term.✓ Proved
\[ = \frac{5 e^{5 x}}{5 x + 1} - \frac{e^{5 x} \frac{d}{d x} \left(5 x + 1\right)}{\left(5 x + 1\right)^{2}} \]
chainApply the chain rule to the second term.✓ Proved
\[ = \frac{5 e^{5 x}}{5 x + 1} - \frac{5 e^{5 x}}{\left(5 x + 1\right)^{2}} \]
derivativealgebraDifferentiate the inner function 5*x + 1. Rewrite the terms using positive exponents.✓ Proved
\[ = \frac{5 \left(5 x + 1\right) e^{5 x} - 5 e^{5 x}}{\left(5 x + 1\right)^{2}} \]
algebraCombine the terms over a common denominator.✓ Proved
\[ = \frac{25 x e^{5 x}}{\left(5 x + 1\right)^{2}} \]
algebrasimplifyDistribute the numerator. Simplify the numerator by canceling terms.✓ Proved
Answer \( \frac{25 x e^{5 x}}{\left(5 x + 1\right)^{2}} \)
Top: the graph of f = e^(5x)/(5x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5e^(5x)/(5x + 1) - 5e^(5x)/(5x + 1)².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{e^{5 x}}{5 x + 1} \]
derivative✓ Proved
\[ = \frac{5 \left(5 x + 1\right) e^{5 x} - 5 e^{5 x}}{\left(5 x + 1\right)^{2}} \]
quotientApply the quotient rule to u = exp(5*x) and v = 5*x + 1✓ Proved
\[ = \frac{25 x e^{5 x}}{\left(5 x + 1\right)^{2}} \]
simplifyCombine terms in the numerator✓ 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
Top: the graph of f = sqrt(2)·sqrt(x)·e^(2x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2sqrt(2)·sqrt(x)·e^(2x) + sqrt(2)·e^(2x)/(2sqrt(x)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
\[ \frac{d}{d x} e^{\left(5 x + 2\right)^{2} + 1} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(\left(5 x + 2\right)^{2} + 1\right) \]
chainApply the chain rule.✓ Proved
\[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(5 x + 2\right)^{2}\right) e^{\left(5 x + 2\right)^{2} + 1} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(5 x + 2\right)^{2} \]
constantalgebraThe derivative of a constant is zero. Simplify the expression.✓ Proved
\[ = \left(10 x + 4\right) e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(5 x + 2\right) \]
powerApply the power rule and chain rule.✓ Proved
\[ = \left(50 x + 20\right) e^{\left(5 x + 2\right)^{2} + 1} \]
derivativeconstant-multiplealgebraDifferentiate the linear term. Multiply the constants. Distribute and simplify the final expression.✓ Proved
Answer \( \left(50 x + 20\right) e^{\left(5 x + 2\right)^{2} + 1} \)
Top: the graph of f = e^((5x + 2)² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (50x + 20)·e^((5x + 2)² + 1).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} e^{\left(5 x + 2\right)^{2} + 1} \]
derivativeStart by identifying the function to differentiate.Not checked
\[ = e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(\left(5 x + 2\right)^{2} + 1\right) \]
chainApply the chain rule: the derivative of exp(u) is exp(u) * u'.✓ Proved
\[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(5 x + 2\right)^{2}\right) e^{\left(5 x + 2\right)^{2} + 1} \]
sumApply the sum rule to the exponent.✓ Proved
\[ = e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(5 x + 2\right)^{2} \]
constantsimplifyThe derivative of the constant 1 is 0. Simplify the expression by removing the zero term.✓ Proved
\[ = \left(10 x + 4\right) e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(5 x + 2\right) \]
chainsimplifyApply the chain rule to the square term: derivative of u^2 is 2u*u'. Simplify the power expression.✓ Proved
sumApply the sum rule to the inner linear term.✓ Proved
\[ = \left(50 x + 20\right) e^{\left(5 x + 2\right)^{2} + 1} \]
constant-multiplesimplifyThe derivative of 5x is 5, and the derivative of constant 2 is 0. Simplify the inner derivative terms.✓ Proved
\[ = 10 \left(5 x + 2\right) e^{\left(5 x + 2\right)^{2} + 1} \]
algebraMultiply the constants 2 and 5.✓ Proved
\[ = \left(50 x + 20\right) e^{\left(5 x + 2\right)^{2} + 1} \]
simplifyRearrange the terms for the final answer.✓ 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
\[ \frac{d}{d x} e^{\left(2 x - 1\right)^{2} + 1} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(\left(2 x - 1\right)^{2} + 1\right) \]
chainApply the chain rule.✓ Proved
\[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(2 x - 1\right)^{2}\right) e^{\left(2 x - 1\right)^{2} + 1} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(2 x - 1\right)^{2} \]
constantalgebraThe derivative of a constant is zero. Simplify the expression by removing the zero term.✓ Proved
\[ = \left(4 x - 2\right) e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(2 x - 1\right) \]
powerApply the power rule and chain rule to the squared term.✓ Proved
\[ = \left(8 x - 4\right) e^{\left(2 x - 1\right)^{2} + 1} \]
derivativealgebrasimplifyDifferentiate the linear term inside. Multiply the constants. Distribute and simplify the final expression.✓ Proved
Answer \( \left(8 x - 4\right) e^{\left(2 x - 1\right)^{2} + 1} \)
Top: the graph of f = e^((2x - 1)² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (8x - 4)·e^((2x - 1)² + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} e^{\left(2 x - 1\right)^{2} + 1} \]
derivative✓ Proved
\[ = e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(\left(2 x - 1\right)^{2} + 1\right) \]
chainApply the chain rule to the outer exponential.✓ Proved
\[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(2 x - 1\right)^{2}\right) e^{\left(2 x - 1\right)^{2} + 1} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = \left(8 x - 4\right) e^{\left(2 x - 1\right)^{2} + 1} \]
powersimplifyDifferentiate (2*x - 1)**2 using the power rule. Combine terms and simplify.✓ 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
simplifyFactor out exp(x-1) and simplify the numerator✓ 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
\[ = \left(x + 2\right)^{2} e^{- x - 2} \frac{d}{d x} \left(- x - 2\right) + e^{- x - 2} \frac{d}{d x} \left(x + 2\right)^{2} \]
chainApply the chain rule to the exponential term.✓ Proved
\[ = - \left(x + 2\right)^{2} e^{- x - 2} + \left(2 x + 4\right) e^{- x - 2} \]
derivativealgebraDifferentiate the inner function and the power term. Simplify the expression.✓ Proved
\[ = - x \left(x + 2\right) e^{- x - 2} \]
algebraalgebraalgebraFactor out the common terms. Simplify the term in the parentheses. Rearrange the terms for the final answer.✓ Proved
Answer \( x \left(- x - 2\right) e^{- x - 2} \)
Top: the graph of f = (x + 2)²·e^(-x - 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(x + 2)²·e^(-x - 2) + (2x + 4)·e^(-x - 2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ = - \left(x + 2\right)^{2} e^{- x - 2} + \left(2 x + 4\right) e^{- x - 2} \]
productpowerexponential✓ Proved
\[ = \left(2 x - \left(x + 2\right)^{2} + 4\right) e^{- x - 2} \]
algebra✓ Proved
\[ = x \left(- x - 2\right) e^{- x - 2} \]
algebra✓ Proved
\[ = - x \left(x + 2\right) e^{- x - 2} \]
simplify✓ 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
powerDifferentiate x^2 using the power rule.✓ Proved
\[ = 2 x + e^{x} + \frac{d}{d x} 4 \]
exponentialDifferentiate e^x, which remains e^x.✓ Proved
\[ = 2 x + e^{x} \]
constantsimplifyThe derivative of the constant 4 is 0. Simplify the expression by removing the zero term.✓ 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
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = 8 x e^{4 x^{2} + 1} \]
powerconstantsimplifyDifferentiate the power term. The derivative of the constant 1 is 0. Simplify the final expression.✓ Proved
Answer \( 8 x e^{4 x^{2} + 1} \)
Top: the graph of f = e^(4x² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 8x·e^(4x² + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} e^{4 x^{2} + 1} \]
derivative✓ Proved
\[ = 8 x e^{4 x^{2} + 1} \]
chainsimplifyApply the chain rule to the outer exponential. Reorder the product for a standard form.✓ 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
\[ \frac{d}{d x} e^{\cos{\left(2 x - 1 \right)}} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = e^{\cos{\left(2 x - 1 \right)}} \frac{d}{d x} \cos{\left(2 x - 1 \right)} \]
chainApply the chain rule to the exponential function.✓ Proved
\[ = - e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right) \]
chainApply the chain rule to the cosine function.✓ Proved
\[ = - e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \frac{d}{d x} 2 x \]
algebraDistribute the derivative to the term 2*x.✓ Proved
\[ = - 2 e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \]
derivativesimplifyDifferentiate the linear term 2*x. Rearrange the terms into a final simplified form.✓ Proved
Answer \( - 2 e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \)
Top: the graph of f = e^(cos(2x - 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2e^(cos(2x - 1))·sin(2x - 1).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} e^{\cos{\left(2 x - 1 \right)}} \]
derivativeStart by identifying the function to differentiate.Not checked
\[ = e^{\cos{\left(2 x - 1 \right)}} \frac{d}{d x} \cos{\left(2 x - 1 \right)} \]
chainApply the chain rule for the outer exponential function: d/dx e^u = e^u * u'.✓ Proved
\[ = - e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right) \]
chainApply the chain rule for the inner cosine function: d/dx cos(v) = -sin(v) * v'.✓ Proved
\[ = - 2 e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \]
constant-multiplesimplifyDifferentiate the linear term 2x - 1 to get 2. Rearrange and simplify the expression by combining constants.✓ 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
\[ \frac{d}{d x} e^{- 3 x} \sin{\left(3 x \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \sin{\left(3 x \right)} \frac{d}{d x} e^{- 3 x} + e^{- 3 x} \frac{d}{d x} \sin{\left(3 x \right)} \]
productApply the product rule.✓ Proved
\[ = - 3 e^{- 3 x} \sin{\left(3 x \right)} + e^{- 3 x} \frac{d}{d x} \sin{\left(3 x \right)} \]
chainDifferentiate the first part using the chain rule.✓ Proved
\[ = - 3 e^{- 3 x} \sin{\left(3 x \right)} + 3 e^{- 3 x} \cos{\left(3 x \right)} \]
chainalgebraDifferentiate the second part using the chain rule. Rearrange the terms.✓ Proved
\[ = 3 \left(- \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{- 3 x} \]
simplifyFactor out the common term.✓ Proved
Answer \( \left(- 3 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\right) e^{- 3 x} \)
Top: the graph of f = e^(-3x)·sin(3x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3e^(-3x)·sin(3x) + 3e^(-3x)·cos(3x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} e^{- 3 x} \sin{\left(3 x \right)} \]
derivative✓ Proved
\[ = \sin{\left(3 x \right)} \frac{d}{d x} e^{- 3 x} + e^{- 3 x} \frac{d}{d x} \sin{\left(3 x \right)} \]
productApply product rule✓ Proved
\[ = - 3 e^{- 3 x} \sin{\left(3 x \right)} + e^{- 3 x} \frac{d}{d x} \sin{\left(3 x \right)} \]
\[ = - 3 e^{- 3 x} \sin{\left(3 x \right)} + 3 e^{- 3 x} \cos{\left(3 x \right)} \]
derivativesimplifyDifferentiate sin(3*x) Factor out 3*exp(-3*x)✓ Proved
\[ = 3 \left(- \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{- 3 x} \]
simplifyFinal simplified derivative✓ 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
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = \left(2 x \frac{d}{d x} x + \frac{d}{d x} 1\right) e^{x^{2} + 1} \]
powerApply the power rule to x**2.✓ Proved
\[ = 2 x e^{x^{2} + 1} \]
derivativesimplifyDifferentiate x and the constant 1. Simplify the expression.✓ Proved
Answer \( 2 x e^{x^{2} + 1} \)
Top: the graph of f = e^(x² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x·e^(x² + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} e^{x^{2} + 1} \]
derivative✓ Proved
\[ = 2 x e^{x^{2} + 1} \]
chainsimplifyApply chain rule to exp(x**2 + 1) Reorder factors for simplicity✓ 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
Top: the graph of f = e^(3 - x)·sin(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -e^(3 - x)·sin(x - 3) + e^(3 - x)·cos(x - 3).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
Top: the graph of f = e^(-x)·sin(x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -e^(-x)·sin(x) + e^(-x)·cos(x).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} e^{- x} \sin{\left(x \right)} \]
derivativeStart by identifying the function to be differentiated.Not checked
algebraFactor out the common term exp(-x).✓ 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
derivativesimplifyDifferentiate log(x). Simplify the expression.✓ 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
Differentiate \( \displaystyle f(x) = e^{\cos{\left(2 x \right)}} \).
\[ \frac{d}{d x} e^{\cos{\left(2 x \right)}} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = e^{\cos{\left(2 x \right)}} \frac{d}{d x} \cos{\left(2 x \right)} \]
chainApply the chain rule to the exponential function.✓ Proved
\[ = - e^{\cos{\left(2 x \right)}} \sin{\left(2 x \right)} \frac{d}{d x} 2 x \]
trigDifferentiate the cosine function.✓ Proved
\[ = - 2 e^{\cos{\left(2 x \right)}} \sin{\left(2 x \right)} \]
derivativesimplifyDifferentiate the inner linear function. Simplify the final expression.✓ Proved
Answer \( - 2 e^{\cos{\left(2 x \right)}} \sin{\left(2 x \right)} \)
Top: the graph of f = e^(cos(2x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2e^(cos(2x))·sin(2x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} e^{\cos{\left(2 x \right)}} \]
derivative✓ Proved
\[ = e^{\cos{\left(2 x \right)}} \frac{d}{d x} \cos{\left(2 x \right)} \]
chainApply chain rule to exp✓ Proved
\[ = - e^{\cos{\left(2 x \right)}} \sin{\left(2 x \right)} \frac{d}{d x} 2 x \]
chainDifferentiate cos(2*x)✓ Proved
\[ = - 2 e^{\cos{\left(2 x \right)}} \sin{\left(2 x \right)} \]
constant-multiplesimplifyDerivative of 2*x is 2 Simplify the product✓ 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
chainsimplifyApply chain rule: derivative of exp(u) is exp(u)*u', u = cos(x) Combine the negative sign✓ 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
More derivatives of exponential functions practice