Home›Calculus 1›Chain rule›Problem 2.907

Derivative of \( \displaystyle \ln{\left(4 x^{2} e^{- 2 x} \right)} \)

Problem 2.907 · hard Beautiful Video

Differentiate \( \displaystyle f(x) = \ln{\left(4 x^{2} e^{- 2 x} \right)} \).

\[ \frac{d}{d x} \ln{\left(4 x^{2} e^{- 2 x} \right)} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(- 2 x + \ln{\left(x^{2} \right)} + \ln{\left(4 \right)}\right) \]
simplifyUse logarithm properties to expand the expression.✓ Proved
\[ = \frac{d}{d x} \left(- 2 x + 2 \ln{\left(x \right)} + \ln{\left(4 \right)}\right) \]
simplifySimplify the logarithmic terms.✓ Proved
\[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \ln{\left(4 \right)} + \frac{d}{d x} 2 \ln{\left(x \right)} \]
sumApply the sum rule for derivatives.✓ Proved
\[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} 2 \ln{\left(x \right)} \]
constantThe derivative of the constant log(4) is zero.✓ Proved
\[ = \frac{d}{d x} \left(- 2 x\right) + 2 \frac{d}{d x} \ln{\left(x \right)} \]
constant-multiplePull out the constant factor 2.✓ Proved
\[ = -2 + \frac{2}{x} \]
derivative algebraDifferentiate the remaining terms. Simplify the expression.✓ Proved

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

Top: the graph of f = log(4x²·e^(-2x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (-8x²·e^(-2x) + 8x·e^(-2x))·e^(2x)/(4x²).

✓ 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	line 1 is the problem as stated
2	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-13: pass — Every line is equal to the previous one and SymPy proved each step. The final answer -2 + 2/x matches. The only issue is style: steps 2 and 3 are labelled 'simplify' where 'logarithmic' would be more precise. The log(x**2) = 2*log(x) step holds only for x>0, but domain restrictions are recorded separately and the derivative is 2/x either way.

agent:pipeline: uphold — Beautiful nomination: The messy log of a product with a Gaussian-style exponential factor splits into log(4) + 2*log(x) - 2*x, so the derivative collapses to the simple -2 + 2/x.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — Every line is equal to the previous one and SymPy proved each step. The final answer -2 + 2/x matches. The only issue is style: steps 2 and 3 are labelled 'simplify' where 'logarithmic' would be more precise. The log(x**2) = 2*log(x) step holds only for x>0, but domain restrictions are recorded separately and the derivative is 2/x either way.
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 3 applies two simplifications (log(x**2)=2*log(x) and log(exp(-2*x))=-2*x) in one step, violating the one‑rule‑per‑step rule. Similarly, step 7 applies two derivative rules simultaneously. These multi‑rule steps make the solution incorrect according to the contract.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by gemma4:26b, checked 2026-09-27 with SymPy 1.14.0.