Home›Calculus 1›Derivatives of exponential functions›Problem 2.638

Derivative of \( \displaystyle \frac{e^{4 x}}{4 x + 1} \)

Problem 2.638 · hard Video

Differentiate \( \displaystyle f(x) = \frac{e^{4 x}}{4 x + 1} \).

\[ \frac{d}{d x} \frac{e^{4 x}}{4 x + 1} \]
rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
\[ = e^{4 x} \frac{d}{d x} \frac{1}{4 x + 1} + \frac{\frac{d}{d x} e^{4 x}}{4 x + 1} \]
productApply the product rule.✓ Proved
\[ = e^{4 x} \frac{d}{d x} \frac{1}{4 x + 1} + \frac{4 e^{4 x}}{4 x + 1} \]
derivativeDifferentiate the first part of the product.✓ Proved
\[ = \frac{4 e^{4 x}}{4 x + 1} - \frac{e^{4 x} \frac{d}{d x} \left(4 x + 1\right)}{\left(4 x + 1\right)^{2}} \]
chainApply the chain rule to the second part.✓ Proved
\[ = \frac{4 e^{4 x}}{4 x + 1} - \frac{4 e^{4 x}}{\left(4 x + 1\right)^{2}} \]
derivative algebraDifferentiate the inner function 4*x + 1. Rewrite the negative exponents as denominators.✓ Proved
\[ = \frac{4 \left(4 x + 1\right) e^{4 x} - 4 e^{4 x}}{\left(4 x + 1\right)^{2}} \]
algebraCombine the terms over a common denominator.✓ Proved
\[ = \frac{16 x e^{4 x}}{\left(4 x + 1\right)^{2}} \]
simplifySimplify the numerator by distributing and canceling terms.✓ Proved

Answer \( \frac{16 x e^{4 x}}{\left(4 x + 1\right)^{2}} \)

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

✓ 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 undefined where 4*x + 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4*x + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4*x + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4*x + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4*x + 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4*x + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4*x + 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4*x + 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 4*x + 1 = 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 the product rule, chain rule, and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, chain rule, and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, chain rule, and algebraic simplification steps. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.
gpt-oss:20b: pass 2026-09-21

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 2026-09-26 with SymPy 1.14.0.