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Logarithmic differentiation practice problems

Functions like xˣ, with the variable in the base and the exponent: take logs first. 50 problems with worked solutions; in 23 of them every equation is proved by a computer algebra system.

13 worked problems

Try each one before opening the solution. Every line is labeled with what checked it; 13 of these also have a second solution, written separately and checked the same way. How we verify

Problem 2.380

hard Video
Differentiate \( \displaystyle f(x) = \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \).
  1. \[ \frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \]
    Start with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    rewriteRewrite the function using the exponential and logarithm form.✓ Proved
  3. \[ = e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \frac{d}{d x} \left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} \]
    chainApply the chain rule for the exponential function.✓ Proved
  4. \[ = \left(\left(5 x + 1\right) \frac{d}{d x} \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} + \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} \frac{d}{d x} \left(5 x + 1\right)\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    productApply the product rule to the exponent.✓ Proved
  5. \[ = \left(\left(5 x + 1\right) \frac{d}{d x} \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    derivativeDifferentiate the first part of the product.✓ Proved
  6. \[ = \left(\frac{\left(5 x + 1\right) \frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = \left(\frac{\left(5 x + 1\right) \left(50 x + 10\right)}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    derivative derivativeDifferentiate the inner function of the logarithm. Differentiate the inner function of the power.✓ Proved
  8. \[ = \left(\frac{10 \left(5 x + 1\right)^{2}}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    algebraSimplify the expression inside the parentheses.✓ Proved
  9. \[ = \left(\frac{10 \left(5 x + 1\right)^{2}}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \]
    simplifyConvert the exponential form back to the original base.✓ Proved
Answer \( 5 \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x} \left(2 \left(5 x + 1\right)^{2} + \left(\left(5 x + 1\right)^{2} + 1\right) \log{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) \)
Top: the graph of f = ((5x + 1)² + 1)^(5x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = ((5x + 1)·(50x + 10)/((5x + 1)² + 1) + 5log((5x + 1)² + 1))·((5x + 1)² + 1)^(5x + 1).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \]
    derivativeStart by defining the derivative of the given function.Not checked
  2. \[ = \frac{d}{d x} e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    rewriteRewrite the base-exponent form using exp and log to handle differentiation.✓ Proved
  3. \[ = e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \frac{d}{d x} \left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} \]
    chainApply the chain rule: derivative of exp(u) is exp(u)*u'.✓ Proved
  4. \[ = \left(\left(5 x + 1\right) \frac{d}{d x} \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} + \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} \frac{d}{d x} \left(5 x + 1\right)\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    productApply the product rule to the exponent term u = (5x+1)log(...).✓ Proved
  5. \[ = \left(\left(5 x + 1\right) \frac{d}{d x} \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    derivativeCompute the derivative of the linear term 5x+1.✓ Proved
  6. \[ = \left(\frac{\left(5 x + 1\right) \frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    chainApply the chain rule for the logarithmic term: derivative of log(v) is v'/v.✓ Proved
  7. \[ = \left(\frac{\left(5 x + 1\right) \left(50 x + 10\right)}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    chainApply the chain rule to the quadratic term inside the log.✓ Proved
  8. \[ = \left(\frac{10 \left(5 x + 1\right)^{2}}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]
    algebra algebraSimplify the constant multiplication in the derivative of the quadratic. Combine the terms in the numerator of the fraction.✓ Proved
  9. \[ = \left(\frac{10 \left(5 x + 1\right)^{2}}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \]
    rewriteRewrite exp(log(...)) back to the original base form.✓ Proved
  10. \[ = \frac{\left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \left(10 \left(5 x + 1\right)^{2} + 5 \left(\left(5 x + 1\right)^{2} + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right)}{\left(5 x + 1\right)^{2} + 1} \]
    algebraFind a common denominator for the terms inside the parentheses.✓ Proved
  11. \[ = \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x} \left(10 \left(5 x + 1\right)^{2} + 5 \left(\left(5 x + 1\right)^{2} + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) \]
    algebraSimplify the power by canceling one factor of the base with the 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

Problem 2.396

hard Video
Differentiate \( \displaystyle f(x) = \left(x^{2} + 1\right)^{x} \).
  1. \[ \frac{d}{d x} \left(x^{2} + 1\right)^{x} \]
    rewriteRewrite the base and exponent using the exponential and logarithm functions.✓ Proved
  2. \[ = \frac{d}{d x} e^{x \ln{\left(x^{2} + 1 \right)}} \]
    chainApply the chain rule to the exponential function.✓ Proved
  3. \[ = e^{x \ln{\left(x^{2} + 1 \right)}} \frac{d}{d x} x \ln{\left(x^{2} + 1 \right)} \]
    productApply the product rule to the exponent.✓ Proved
  4. \[ = \left(x \frac{d}{d x} \ln{\left(x^{2} + 1 \right)} + \ln{\left(x^{2} + 1 \right)} \frac{d}{d x} x\right) e^{x \ln{\left(x^{2} + 1 \right)}} \]
    derivativeDifferentiate the first part of the product.✓ Proved
  5. \[ = \left(x \frac{d}{d x} \ln{\left(x^{2} + 1 \right)} + \ln{\left(x^{2} + 1 \right)}\right) e^{x \ln{\left(x^{2} + 1 \right)}} \]
    derivativeDifferentiate the second part of the product.✓ Proved
  6. \[ = \left(\frac{x \frac{d}{d x} \left(x^{2} + 1\right)}{x^{2} + 1} + \ln{\left(x^{2} + 1 \right)}\right) e^{x \ln{\left(x^{2} + 1 \right)}} \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = \left(\frac{2 x^{2}}{x^{2} + 1} + \ln{\left(x^{2} + 1 \right)}\right) e^{x \ln{\left(x^{2} + 1 \right)}} \]
    derivative algebraDifferentiate the inner function of the logarithm. Simplify the expression inside the parentheses.✓ Proved
  8. \[ = \left(x^{2} + 1\right)^{x} \left(\frac{2 x^{2}}{x^{2} + 1} + \ln{\left(x^{2} + 1 \right)}\right) \]
    rewrite simplifyConvert the exponential form back to the original power form. Final simplification.✓ Proved
Answer \( \left(x^{2} + 1\right)^{x} \left(\frac{2 x^{2}}{x^{2} + 1} + \log{\left(x^{2} + 1 \right)}\right) \)
Top: the graph of f = (x² + 1)^x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (x² + 1)^x·(2x²/(x² + 1) + log(x² + 1)).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(x^{2} + 1\right)^{x} \]
    derivativeStart with the derivative of the given function.Not checked
  2. \[ = \frac{d}{d x} e^{x \ln{\left(x^{2} + 1 \right)}} \]
    rewriteRewrite the base-exponent form using the identity a^b = exp(b*log(a)).✓ Proved
  3. \[ = e^{x \ln{\left(x^{2} + 1 \right)}} \frac{d}{d x} x \ln{\left(x^{2} + 1 \right)} \]
    chainApply the chain rule for the exponential function: d/dx exp(u) = exp(u) * u'.✓ Proved
  4. \[ = \left(x \frac{d}{d x} \ln{\left(x^{2} + 1 \right)} + \ln{\left(x^{2} + 1 \right)} \frac{d}{d x} x\right) e^{x \ln{\left(x^{2} + 1 \right)}} \]
    productApply the product rule to the exponent u = x * log(x^2 + 1).✓ Proved
  5. \[ = \left(x \frac{d}{d x} \ln{\left(x^{2} + 1 \right)} + \ln{\left(x^{2} + 1 \right)}\right) e^{x \ln{\left(x^{2} + 1 \right)}} \]
    derivativeDifferentiate the first term x in the product to get 1.✓ Proved
  6. \[ = \left(\frac{x \frac{d}{d x} \left(x^{2} + 1\right)}{x^{2} + 1} + \ln{\left(x^{2} + 1 \right)}\right) e^{x \ln{\left(x^{2} + 1 \right)}} \]
    chainApply the chain rule for the logarithm: d/dx log(v) = v'/v.✓ Proved
  7. \[ = \left(\frac{2 x^{2}}{x^{2} + 1} + \ln{\left(x^{2} + 1 \right)}\right) e^{x \ln{\left(x^{2} + 1 \right)}} \]
    derivative algebraDifferentiate the inner function x^2 + 1 to get 2x. Simplify the term x * (2x / (x^2 + 1)) to 2x^2 / (x^2 + 1).✓ Proved
  8. \[ = \left(x^{2} + 1\right)^{x} \left(\frac{2 x^{2}}{x^{2} + 1} + \ln{\left(x^{2} + 1 \right)}\right) \]
    rewriteRewrite exp(x * log(x^2 + 1)) back to (x^2 + 1)^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

Problem 2.527

hard Video
Differentiate \( \displaystyle f(x) = \left(\left(x - 3\right)^{2} + 1\right)^{x - 3} \).
  1. \[ \frac{d}{d x} \left(\left(x - 3\right)^{2} + 1\right)^{x - 3} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    rewriteRewrite the function using the exponential identity.✓ Proved
  3. \[ = e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \frac{d}{d x} \left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)} \]
    chainApply the chain rule.✓ Proved
  4. \[ = \left(\left(x - 3\right) \frac{d}{d x} \ln{\left(\left(x - 3\right)^{2} + 1 \right)} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)} \frac{d}{d x} \left(x - 3\right)\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    productApply the product rule to the inner expression.✓ Proved
  5. \[ = \left(\left(x - 3\right) \frac{d}{d x} \ln{\left(\left(x - 3\right)^{2} + 1 \right)} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    derivativeDifferentiate the first part of the product.✓ Proved
  6. \[ = \left(\frac{\left(x - 3\right) \frac{d}{d x} \left(\left(x - 3\right)^{2} + 1\right)}{\left(x - 3\right)^{2} + 1} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = \left(\frac{\left(x - 3\right) \left(2 x - 6\right)}{\left(x - 3\right)^{2} + 1} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    derivativeDifferentiate the inner quadratic expression.✓ Proved
  8. \[ = \left(\frac{2 \left(x - 3\right)^{2}}{\left(x - 3\right)^{2} + 1} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    algebraSimplify the product in the second term.✓ Proved
  9. \[ = \left(\frac{2 \left(x - 3\right)^{2}}{\left(x - 3\right)^{2} + 1} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) \left(\left(x - 3\right)^{2} + 1\right)^{x - 3} \]
    simplifyConvert the exponential form back to the original base.✓ Proved
Answer \( \left(\left(x - 3\right)^{2} + 1\right)^{x - 4} \left(2 \left(x - 3\right)^{2} + \left(\left(x - 3\right)^{2} + 1\right) \log{\left(\left(x - 3\right)^{2} + 1 \right)}\right) \)
Top: the graph of f = ((x - 3)² + 1)^(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = ((x - 3)·(2x - 6)/((x - 3)² + 1) + log((x - 3)² + 1))·((x - 3)² + 1)^(x - 3).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\left(x - 3\right)^{2} + 1\right)^{x - 3} \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    rewrite✓ Proved
  3. \[ = e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \frac{d}{d x} \left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)} \]
    chain✓ Proved
  4. \[ = \left(\left(x - 3\right) \frac{d}{d x} \ln{\left(\left(x - 3\right)^{2} + 1 \right)} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)} \frac{d}{d x} \left(x - 3\right)\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    product✓ Proved
  5. \[ = \left(\left(x - 3\right) \frac{d}{d x} \ln{\left(\left(x - 3\right)^{2} + 1 \right)} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    constant-multiple simplify✓ Proved
  6. \[ = \left(\frac{\left(x - 3\right) \frac{d}{d x} \left(\left(x - 3\right)^{2} + 1\right)}{\left(x - 3\right)^{2} + 1} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    chain✓ Proved
  7. \[ = \left(\frac{\left(x - 3\right) \frac{d}{d x} \left(x - 3\right)^{2}}{\left(x - 3\right)^{2} + 1} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    simplify✓ Proved
  8. \[ = \left(\frac{2 \left(x - 3\right)^{2}}{\left(x - 3\right)^{2} + 1} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) e^{\left(x - 3\right) \ln{\left(\left(x - 3\right)^{2} + 1 \right)}} \]
    power simplify✓ Proved
  9. \[ = \left(\frac{2 \left(x - 3\right)^{2}}{\left(x - 3\right)^{2} + 1} + \ln{\left(\left(x - 3\right)^{2} + 1 \right)}\right) \left(\left(x - 3\right)^{2} + 1\right)^{x - 3} \]
    rewrite✓ 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 2.651

hard Video
Differentiate \( \displaystyle f(x) = \left(\left(x + 2\right)^{2} + 1\right)^{x + 2} \).
  1. \[ \frac{d}{d x} \left(\left(x + 2\right)^{2} + 1\right)^{x + 2} \]
    Start with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    rewriteRewrite the function using the exponential and logarithm identity.✓ Proved
  3. \[ = e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \frac{d}{d x} \left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)} \]
    chainApply the chain rule for the exponential function.✓ Proved
  4. \[ = \left(\left(x + 2\right) \frac{d}{d x} \ln{\left(\left(x + 2\right)^{2} + 1 \right)} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)} \frac{d}{d x} \left(x + 2\right)\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    productApply the product rule to the exponent's derivative.✓ Proved
  5. \[ = \left(\left(x + 2\right) \frac{d}{d x} \ln{\left(\left(x + 2\right)^{2} + 1 \right)} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    derivativeDifferentiate the first part of the product.✓ Proved
  6. \[ = \left(\frac{\left(x + 2\right) \frac{d}{d x} \left(\left(x + 2\right)^{2} + 1\right)}{\left(x + 2\right)^{2} + 1} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = \left(\frac{\left(x + 2\right) \left(2 x + 4\right)}{\left(x + 2\right)^{2} + 1} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    chainDifferentiate the inner function of the logarithm.✓ Proved
  8. \[ = \left(\frac{2 \left(x + 2\right)^{2}}{\left(x + 2\right)^{2} + 1} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    algebraSimplify the expression algebraically.✓ Proved
  9. \[ = \left(\frac{2 \left(x + 2\right)^{2}}{\left(x + 2\right)^{2} + 1} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) \left(\left(x + 2\right)^{2} + 1\right)^{x + 2} \]
    simplifySubstitute back the original base for the final simplified form.✓ Proved
Answer \( \left(\left(x + 2\right)^{2} + 1\right)^{x + 1} \left(2 \left(x + 2\right)^{2} + \left(\left(x + 2\right)^{2} + 1\right) \log{\left(\left(x + 2\right)^{2} + 1 \right)}\right) \)
Top: the graph of f = ((x + 2)² + 1)^(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = ((x + 2)·(2x + 4)/((x + 2)² + 1) + log((x + 2)² + 1))·((x + 2)² + 1)^(x + 2).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\left(x + 2\right)^{2} + 1\right)^{x + 2} \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    rewrite✓ Proved
  3. \[ = e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \frac{d}{d x} \left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)} \]
    chain✓ Proved
  4. \[ = \left(\left(x + 2\right) \frac{d}{d x} \ln{\left(\left(x + 2\right)^{2} + 1 \right)} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)} \frac{d}{d x} \left(x + 2\right)\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    product✓ Proved
  5. \[ = \left(\left(x + 2\right) \frac{d}{d x} \ln{\left(\left(x + 2\right)^{2} + 1 \right)} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    constant-multiple simplify✓ Proved
  6. \[ = \left(\frac{\left(x + 2\right) \frac{d}{d x} \left(\left(x + 2\right)^{2} + 1\right)}{\left(x + 2\right)^{2} + 1} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    logarithmic✓ Proved
  7. \[ = \left(\frac{\left(x + 2\right) \frac{d}{d x} \left(x + 2\right)^{2}}{\left(x + 2\right)^{2} + 1} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    simplify✓ Proved
  8. \[ = \left(\frac{2 \left(x + 2\right)^{2}}{\left(x + 2\right)^{2} + 1} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) e^{\left(x + 2\right) \ln{\left(\left(x + 2\right)^{2} + 1 \right)}} \]
    power simplify✓ Proved
  9. \[ = \left(\frac{2 \left(x + 2\right)^{2}}{\left(x + 2\right)^{2} + 1} + \ln{\left(\left(x + 2\right)^{2} + 1 \right)}\right) \left(\left(x + 2\right)^{2} + 1\right)^{x + 2} \]
    rewrite✓ 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 2.470

hard Video
Differentiate \( \displaystyle f(x) = e^{\left(x - 3\right)^{2}} \).
  1. \[ \frac{d}{d x} e^{\left(x - 3\right)^{2}} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = e^{\left(x - 3\right)^{2}} \frac{d}{d x} \left(x - 3\right)^{2} \]
    chainApply the chain rule.✓ Proved
  3. \[ = \left(2 x - 6\right) e^{\left(x - 3\right)^{2}} \frac{d}{d x} \left(x - 3\right) \]
    powerApply the power rule to the inner function.✓ Proved
  4. \[ = \left(2 x - 6\right) e^{\left(x - 3\right)^{2}} \]
    derivative algebra algebraDifferentiate the inner linear term. Distribute the 2. Rearrange the terms.✓ Proved
Answer \( \left(2 x - 6\right) e^{\left(x - 3\right)^{2}} \)
Top: the graph of f = e^((x - 3)²). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (2x - 6)·e^((x - 3)²).
A second solution, written separately

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

  1. \[ \frac{d}{d x} e^{\left(x - 3\right)^{2}} \]
    derivativeStart by identifying the function to differentiate.Not checked
  2. \[ = e^{\left(x - 3\right)^{2}} \frac{d}{d x} \left(x - 3\right)^{2} \]
    chainApply the chain rule: the derivative of exp(u) is exp(u) times the derivative of u.✓ Proved
  3. \[ = e^{\left(x - 3\right)^{2}} \frac{d}{d x} \left(x^{2} - 6 x + 9\right) \]
    algebraExpand the exponent (x-3)^2 to x^2 - 6x + 9 for easier differentiation.✓ Proved
  4. \[ = \left(\frac{d}{d x} 9 + \frac{d}{d x} \left(- 6 x\right) + \frac{d}{d x} x^{2}\right) e^{\left(x - 3\right)^{2}} \]
    sumApply the sum rule to differentiate each term in the polynomial.✓ Proved
  5. \[ = \left(2 x - 6\right) e^{\left(x - 3\right)^{2}} \]
    power simplifyDifferentiate x^2 to get 2x, -6x to get -6, and the constant 9 to get 0. Simplify the expression inside the parentheses 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

Problem 2.620

hard Video
Differentiate \( \displaystyle f(x) = e^{\left(2 x + 1\right)^{2}} \).
  1. \[ \frac{d}{d x} e^{\left(2 x + 1\right)^{2}} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = e^{\left(2 x + 1\right)^{2}} \frac{d}{d x} \left(2 x + 1\right)^{2} \]
    chainApply the chain rule.✓ Proved
  3. \[ = \left(4 x + 2\right) e^{\left(2 x + 1\right)^{2}} \frac{d}{d x} \left(2 x + 1\right) \]
    powerApply the power rule to the exponent.✓ Proved
  4. \[ = \left(8 x + 4\right) e^{\left(2 x + 1\right)^{2}} \]
    derivative algebra algebraDifferentiate the inner linear function. Simplify the constants. Rearrange the terms.✓ Proved
Answer \( \left(8 x + 4\right) e^{\left(2 x + 1\right)^{2}} \)
Top: the graph of f = e^((2x + 1)²). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (8x + 4)·e^((2x + 1)²).
A second solution, written separately

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

  1. \[ \frac{d}{d x} e^{\left(2 x + 1\right)^{2}} \]
    derivative✓ Proved
  2. \[ = e^{\left(2 x + 1\right)^{2}} \frac{d}{d x} \left(2 x + 1\right)^{2} \]
    chainApply chain rule to exp(u).✓ Proved
  3. \[ = 4 \left(2 x + 1\right) e^{\left(2 x + 1\right)^{2}} \]
    powerDifferentiate (2*x + 1)**2.✓ Proved
  4. \[ = \left(8 x + 4\right) e^{\left(2 x + 1\right)^{2}} \]
    algebra simplifyReorder factors. Simplify 4*(2*x + 1).✓ 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 2.519

hard Video
Differentiate \( \displaystyle f(x) = e^{16 x^{2}} \).
  1. \[ \frac{d}{d x} e^{16 x^{2}} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = e^{16 x^{2}} \frac{d}{d x} 16 x^{2} \]
    chainApply the chain rule.✓ Proved
  3. \[ = 16 e^{16 x^{2}} \frac{d}{d x} x^{2} \]
    constant-multiplePull out the constant factor.✓ Proved
  4. \[ = 32 x e^{16 x^{2}} \]
    power simplifyDifferentiate the power function. Multiply the terms to get the final result.✓ Proved
Answer \( 32 x e^{16 x^{2}} \)
Top: the graph of f = e^(16x²). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 32x·e^(16x²).
A second solution, written separately

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

  1. \[ \frac{d}{d x} e^{16 x^{2}} \]
    derivativeStart by identifying the function to differentiate.Not checked
  2. \[ = e^{16 x^{2}} \frac{d}{d x} 16 x^{2} \]
    chainApply the chain rule: the derivative of exp(u) is exp(u) * u'.✓ Proved
  3. \[ = 32 x e^{16 x^{2}} \]
    power algebraDifferentiate the inner function 16*x^2 using the power rule. Rearrange the terms for the final simplified 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

Problem 2.1054

hard Video
Differentiate \( \displaystyle f(x) = e^{x^{2}} \).
  1. \[ \frac{d}{d x} e^{x^{2}} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = e^{x^{2}} \frac{d}{d x} x^{2} \]
    chainApply the chain rule.✓ Proved
  3. \[ = 2 x e^{x^{2}} \]
    derivative algebraDifferentiate the inner function x**2. Rearrange the terms for the final answer.✓ Proved
Answer \( 2 x e^{x^{2}} \)
Top: the graph of f = e^(x²). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x·e^(x²).
A second solution, written separately

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

  1. \[ \frac{d}{d x} e^{x^{2}} \]
    derivative✓ Proved
  2. \[ = e^{x^{2}} \frac{d}{d x} x^{2} \]
    chain✓ Proved
  3. \[ = 2 x e^{x^{2}} \]
    derivative 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

Problem 2.338

medium Video
Differentiate \( \displaystyle f(x) = x^{x - 1} \).
  1. \[ \frac{d}{d x} x^{x - 1} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} e^{\left(x - 1\right) \ln{\left(x \right)}} \]
    rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
  3. \[ = e^{\left(x - 1\right) \ln{\left(x \right)}} \frac{d}{d x} \left(x - 1\right) \ln{\left(x \right)} \]
    chainApply the chain rule.✓ Proved
  4. \[ = \left(\frac{d}{d x} x \ln{\left(x \right)} - \frac{d}{d x} \ln{\left(x \right)}\right) e^{\left(x - 1\right) \ln{\left(x \right)}} \]
    algebraDistribute the derivative over the subtraction.✓ Proved
  5. \[ = \left(\frac{d}{d x} x \ln{\left(x \right)} - \frac{1}{x}\right) e^{\left(x - 1\right) \ln{\left(x \right)}} \]
    derivativeDifferentiate the log(x) term.✓ Proved
  6. \[ = \left(x \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} x - \frac{1}{x}\right) e^{\left(x - 1\right) \ln{\left(x \right)}} \]
    productApply the product rule to x * log(x).✓ Proved
  7. \[ = \left(\ln{\left(x \right)} + 1 - \frac{1}{x}\right) e^{\left(x - 1\right) \ln{\left(x \right)}} \]
    derivative algebraDifferentiate the individual terms. Simplify the expression inside the parentheses.✓ Proved
  8. \[ = x^{x - 1} \left(\ln{\left(x \right)} + 1 - \frac{1}{x}\right) \]
    simplifyConvert the exponential back to the original power form.≈ Checked numerically
Answer \( x^{x - 1} \left(\log{\left(x \right)} + \frac{x - 1}{x}\right) \)
Top: the graph of f = x^(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x^(x - 1)·(log(x) + (x - 1)/x).
A second solution, written separately

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

  1. \[ \frac{d}{d x} x^{x - 1} \]
    derivative✓ Proved
  2. \[ = x^{x - 1} \left(\ln{\left(x \right)} + \frac{x - 1}{x}\right) \]
    logarithmicApply logarithmic differentiation to x**(x-1).✓ Proved
  3. \[ = x^{x - 1} \left(\ln{\left(x \right)} + 1 - \frac{1}{x}\right) \]
    simplifySimplify the fraction (x-1)/x to 1-1/x.✓ Proved

✓ Nihil obstat Lines: 8 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 2.366

medium Video
Differentiate \( \displaystyle f(x) = \left(2 x + 1\right)^{2 x} \).
  1. \[ \frac{d}{d x} \left(2 x + 1\right)^{2 x} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} e^{2 x \ln{\left(2 x + 1 \right)}} \]
    rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
  3. \[ = e^{2 x \ln{\left(2 x + 1 \right)}} \frac{d}{d x} 2 x \ln{\left(2 x + 1 \right)} \]
    chainApply the chain rule for the exponential function.✓ Proved
  4. \[ = \left(2 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} 2 x\right) e^{2 x \ln{\left(2 x + 1 \right)}} \]
    productApply the product rule to the inner expression.✓ Proved
  5. \[ = \left(2 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 2 \ln{\left(2 x + 1 \right)}\right) e^{2 x \ln{\left(2 x + 1 \right)}} \]
    derivativeDifferentiate 2*x.✓ Proved
  6. \[ = \left(\frac{2 x \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} + 2 \ln{\left(2 x + 1 \right)}\right) e^{2 x \ln{\left(2 x + 1 \right)}} \]
    derivativeDifferentiate the logarithm using the chain rule.✓ Proved
  7. \[ = \left(\frac{4 x}{2 x + 1} + 2 \ln{\left(2 x + 1 \right)}\right) e^{2 x \ln{\left(2 x + 1 \right)}} \]
    derivative algebraDifferentiate 2*x + 1. Simplify the expression inside the parentheses.✓ Proved
  8. \[ = \left(2 x + 1\right)^{2 x} \left(\frac{4 x}{2 x + 1} + 2 \ln{\left(2 x + 1 \right)}\right) \]
    simplifyConvert the exponential form back to the original power form.≈ Checked numerically
Answer \( \left(2 x + 1\right)^{2 x} \left(\frac{4 x}{2 x + 1} + 2 \log{\left(2 x + 1 \right)}\right) \)
Top: the graph of f = (2x + 1)^(2x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (2x + 1)^(2x)·(4x/(2x + 1) + 2log(2x + 1)).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(2 x + 1\right)^{2 x} \]
    derivative✓ Proved
  2. \[ = \left(2 x + 1\right)^{2 x} \left(\frac{4 x}{2 x + 1} + 2 \ln{\left(2 x + 1 \right)}\right) \]
    powerApply the general power rule for a(x)**b(x).✓ Proved

✓ Nihil obstat Lines: 8 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 2.371

medium Video
Differentiate \( \displaystyle f(x) = \left(x + 2\right)^{x + 1} \).
  1. \[ \frac{d}{d x} \left(x + 2\right)^{x + 1} \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} e^{\left(x + 1\right) \ln{\left(x + 2 \right)}} \]
    rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
  3. \[ = e^{\left(x + 1\right) \ln{\left(x + 2 \right)}} \frac{d}{d x} \left(x + 1\right) \ln{\left(x + 2 \right)} \]
    chainApply the chain rule.✓ Proved
  4. \[ = \left(\left(x + 1\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} \left(x + 1\right)\right) e^{\left(x + 1\right) \ln{\left(x + 2 \right)}} \]
    productApply the product rule to the inner expression.✓ Proved
  5. \[ = \left(\left(x + 1\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)}\right) e^{\left(x + 1\right) \ln{\left(x + 2 \right)}} \]
    derivativeDifferentiate x + 1.✓ Proved
  6. \[ = \left(\frac{x + 1}{x + 2} + \ln{\left(x + 2 \right)}\right) e^{\left(x + 1\right) \ln{\left(x + 2 \right)}} \]
    derivative simplifyDifferentiate log(x + 2). Simplify the expression.✓ Proved
  7. \[ = \left(x + 2\right)^{x + 1} \left(\frac{x + 1}{x + 2} + \ln{\left(x + 2 \right)}\right) \]
    simplifyConvert the exponential form back to the original power form.≈ Checked numerically
Answer \( \left(x + 2\right)^{x} \left(x + \left(x + 2\right) \log{\left(x + 2 \right)} + 1\right) \)
Top: the graph of f = (x + 2)^(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (x + 2)^(x + 1)·((x + 1)/(x + 2) + log(x + 2)).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(x + 2\right)^{x + 1} \]
    derivative✓ Proved
  2. \[ = \left(x + 2\right)^{x + 1} \left(\frac{x + 1}{x + 2} + \ln{\left(x + 2 \right)}\right) \]
    power simplify simplifyApply the general power rule for a^b Differentiate the exponents and base: (x+1)'=1, (x+2)'=1 Combine constants and simplify the fraction✓ Proved

✓ Nihil obstat Lines: 7 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 2.819

medium Video
Differentiate \( \displaystyle f(x) = \left(x + 1\right)^{x} \).
  1. \[ \frac{d}{d x} \left(x + 1\right)^{x} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} e^{x \ln{\left(x + 1 \right)}} \]
    rewriteRewrite the function using the exponential and logarithm base change.≈ Checked numerically
  3. \[ = e^{x \ln{\left(x + 1 \right)}} \frac{d}{d x} x \ln{\left(x + 1 \right)} \]
    chainApply the chain rule for the exponential function.✓ Proved
  4. \[ = \left(x \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)} \frac{d}{d x} x\right) e^{x \ln{\left(x + 1 \right)}} \]
    productApply the product rule to the exponent expression.✓ Proved
  5. \[ = \left(x \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)}\right) e^{x \ln{\left(x + 1 \right)}} \]
    constant-multipleSimplify the derivative of x.✓ Proved
  6. \[ = \left(\frac{x}{x + 1} + \ln{\left(x + 1 \right)}\right) e^{x \ln{\left(x + 1 \right)}} \]
    logarithmic algebraDifferentiate the natural logarithm. Simplify the expression.✓ Proved
  7. \[ = \left(x + 1\right)^{x} \left(\frac{x}{x + 1} + \ln{\left(x + 1 \right)}\right) \]
    simplifyConvert back to the original base form.≈ Checked numerically
Answer \( \left(x + 1\right)^{x} \left(\frac{x}{x + 1} + \ln{\left(x + 1 \right)}\right) \)
Top: the graph of f = (x + 1)^x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (x + 1)^x·(x/(x + 1) + log(x + 1)).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(x + 1\right)^{x} \]
    derivative✓ Proved
  2. \[ = \left(x + 1\right)^{x} \left(x \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)} \frac{d}{d x} x\right) \]
    powerApply power rule for variable exponent✓ Proved
  3. \[ = \left(x + 1\right)^{x} \left(\frac{x}{x + 1} + \ln{\left(x + 1 \right)}\right) \]
    derivative simplifyDifferentiate x and log(x+1) Simplify the expression✓ Proved

✓ Nihil obstat Lines: 7 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 2.1280

medium Video
Differentiate \( \displaystyle f(x) = \left(x + 1\right)^{x + 1} \).
  1. \[ \frac{d}{d x} \left(x + 1\right)^{x + 1} \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} e^{\left(x + 1\right) \ln{\left(x + 1 \right)}} \]
    rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
  3. \[ = e^{\left(x + 1\right) \ln{\left(x + 1 \right)}} \frac{d}{d x} \left(x + 1\right) \ln{\left(x + 1 \right)} \]
    chainApply the chain rule.✓ Proved
  4. \[ = \left(\left(x + 1\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right)\right) e^{\left(x + 1\right) \ln{\left(x + 1 \right)}} \]
    productApply the product rule to the inner term.✓ Proved
  5. \[ = \left(\left(x + 1\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)}\right) e^{\left(x + 1\right) \ln{\left(x + 1 \right)}} \]
    derivativeDifferentiate the first factor of the product.✓ Proved
  6. \[ = \left(\ln{\left(x + 1 \right)} + 1\right) e^{\left(x + 1\right) \ln{\left(x + 1 \right)}} \]
    derivative algebraDifferentiate the second factor of the product. Simplify the expression inside the parentheses.✓ Proved
  7. \[ = \left(x + 1\right)^{x + 1} \left(\ln{\left(x + 1 \right)} + 1\right) \]
    simplifyConvert the exponential form back to the original power form.≈ Checked numerically
Answer \( \left(x + 1\right)^{x + 1} \left(\ln{\left(x + 1 \right)} + 1\right) \)
Top: the graph of f = (x + 1)^(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (x + 1)^(x + 1)·(log(x + 1) + 1).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(x + 1\right)^{x + 1} \]
    derivative✓ Proved
  2. \[ = \left(x + 1\right)^{x + 1} \left(\ln{\left(x + 1 \right)} + 1\right) \]
    power✓ Proved

✓ Nihil obstat Lines: 7 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

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Problem 2.562hard Video✓ Checked independently
Differentiate \( \displaystyle f(x) = \sin^{x}{\left(x \right)} \).
Problem 2.581hard Video✓ Checked independently
Differentiate \( \displaystyle f(x) = \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \).
Problem 2.597hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin^{4 x}{\left(4 x \right)} \).
Problem 2.602hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - 3\right)^{\sqrt{x - 3}} \).
Problem 2.825hard Video✓ Checked independently
Differentiate \( \displaystyle f(x) = \sin^{4 x - 3}{\left(4 x - 3 \right)} \).
Problem 2.834hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x + 1 \right)}^{x + 1} \).
Problem 2.841hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{\frac{1}{x}} \).
Problem 2.936hard Video≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(4 x + 2\right)^{\sin{\left(4 x + 2 \right)}} \).
Problem 2.937hard Video≈ Checked numerically
Differentiate \( \displaystyle f(x) = x^{\sqrt{x}} \).
Problem 2.939hard Video≈ Checked numerically
Differentiate \( \displaystyle f(x) = x^{x^{2}} \).
Problem 2.941hard Video≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(2 x\right)^{\cos{\left(2 x \right)}} \).
Problem 2.1053hard Video✓ Nihil obstat