Home›Calculus 1›Derivatives of logarithms›Problem 2.2295

Derivative of \( \displaystyle x - \left(x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)} \)

Problem 2.2295 · hard Video

Differentiate \( \displaystyle f(x) = x - \left(x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)} \).

\[ \frac{d}{d x} \left(x - \left(x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} x - \frac{d}{d x} \left(x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)} \]
sumApply the difference rule.✓ Proved
\[ = - \left(x + \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} - \ln{\left(5 x + 1 \right)} \frac{d}{d x} \left(x + \frac{1}{5}\right) + \frac{d}{d x} x \]
productApply the product rule to the second term.✓ Proved
\[ = - \left(x + \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} - \ln{\left(5 x + 1 \right)} \frac{d}{d x} \frac{1}{5} - \ln{\left(5 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
sumDistribute the derivative over the sum in the product rule.✓ Proved
\[ = - \left(x + \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} - \ln{\left(5 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
derivative simplifyDifferentiate the constant term 1/5. Simplify the expression by removing the zero term.✓ Proved
\[ = - \left(x + \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} - \ln{\left(5 x + 1 \right)} + 1 \]
derivative derivativeDifferentiate x and the product term. Differentiate x.✓ Proved
\[ = - \frac{\left(x + \frac{1}{5}\right) \frac{d}{d x} \left(5 x + 1\right)}{5 x + 1} - \ln{\left(5 x + 1 \right)} + 1 \]
logarithmicApply the chain rule for the logarithm.✓ Proved
\[ = - \frac{5 \left(x + \frac{1}{5}\right)}{5 x + 1} - \ln{\left(5 x + 1 \right)} + 1 \]
derivativeDifferentiate the inner function 5*x + 1.✓ Proved
\[ = - \ln{\left(5 x + 1 \right)} \]
algebra algebra algebra simplifySimplify the fraction (x + 1/5)*5. Simplify the fraction to 1. Distribute the negative sign. Final simplification.✓ Proved

Answer \( - \ln{\left(5 x + 1 \right)} \)

Top: the graph of f = x - (x + 1/5)·log(5x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -5(x + 1/5)/(5x + 1) - log(5x + 1) + 1.

✓ 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
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 undefined where 5*x + 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments
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: fail (style) — Step 4 violates the one-rule-per-step constraint by applying the sum rule to expand the derivative of a sum inside the product rule expansion simultaneously. Step 9 is mislabeled as 'logarithmic' when it applies the chain rule to the logarithm; the label 'chain' is required for the inner derivative.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — Step 4 violates the one-rule-per-step constraint by applying the sum rule to expand the derivative of a sum inside the product rule expansion simultaneously. Step 9 is mislabeled as 'logarithmic' when it applies the chain rule to the logarithm; the label 'chain' is required for the inner derivative.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (style) 2026-10-15 — Step 7 applies the derivative rule to two distinct terms (x and the product term) simultaneously, violating the one-change-per-step constraint. Step 8 is redundant and incorrectly labeled as applying the derivative rule when it merely simplifies the coefficient 1.
gpt-oss:20b: fail (style) 2026-10-15 — Step 8 incorrectly labels an algebraic simplification as a derivative. The step merely replaces 1*log(5*x+1) with log(5*x+1); it should be labeled "algebra" or "simplify" rather than "derivative".
qwen3.6:27b-mlx: fail (style) 2026-10-14 — Step 4 applies the sum rule to expand Derivative(x + 1/5, x) into Derivative(x, x) + Derivative(1/5, x), but it also distributes the outer terms, effectively applying the product rule expansion and the sum rule simultaneously. Step 9 is labeled 'logarithmic' but explicitly notes applying the chain rule; while 'logarithmic' is acceptable for the log derivative, the note is slightly misleading about the rule name used in the label.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.