Home›Calculus 1›Chain rule›Problem 2.1143

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

Problem 2.1143 · hard Beautiful Video

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

\[ \frac{d}{d x} \ln{\left(\frac{e^{2 x}}{4 x^{2}} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(2 x - \ln{\left(4 x^{2} \right)}\right) \]
algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(2*x)) to 2*x.✓ Proved
\[ = \frac{d}{d x} 2 x - \frac{d}{d x} \ln{\left(4 x^{2} \right)} \]
sumSplit the derivative using the difference rule.✓ Proved
\[ = 2 - \frac{d}{d x} \ln{\left(4 x^{2} \right)} \]
constantThe derivative of 2*x is 2.✓ Proved
\[ = 2 - \frac{d}{d x} \left(\ln{\left(x^{2} \right)} + \ln{\left(4 \right)}\right) \]
algebraUse the property log(ab) = log(a) + log(b).✓ Proved
\[ = - \frac{d}{d x} \ln{\left(4 \right)} - \frac{d}{d x} \ln{\left(x^{2} \right)} + 2 \]
sumDistribute the derivative.✓ Proved
\[ = 2 - \frac{d}{d x} \ln{\left(x^{2} \right)} \]
constantThe derivative of the constant log(4) is 0.✓ Proved
\[ = 2 - \frac{d}{d x} 2 \ln{\left(x \right)} \]
rewriteUse the power rule for logarithms: log(x**2) = 2*log(x).✓ Proved
\[ = 2 - 2 \frac{d}{d x} \ln{\left(x \right)} \]
constant-multiplePull the constant factor 2 out of the derivative.✓ Proved
\[ = 2 - \frac{2}{x} \]
derivative simplifyThe derivative of log(x) is 1/x. Simplify the expression.✓ Proved

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

Top: the graph of f = log(e^(2x)/(4x²)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4x²·(e^(2x)/(2x²) - e^(2x)/(2x³))·e^(-2x).

✓ 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 undefined where x = 0
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
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
12	✓ 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 — The solution correctly applies one rule per step, uses valid labels from the vocabulary, and provides accurate notes. The algebraic simplifications and differentiation rules are applied correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies one rule per step, uses valid labels from the vocabulary, and provides accurate notes. The algebraic simplifications and differentiation rules are applied correctly.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies logarithmic properties to simplify the expression before differentiating, and each step adheres to the single-rule constraint with appropriate labels.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28
gpt-oss:20b: pass 2026-09-28

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 2026-10-03 with SymPy 1.14.0.