Home›Calculus 1›Derivatives of logarithms›Problem 2.2272

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

Problem 2.2272 · hard Beautiful 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
\[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \left(x + \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} \left(x + \frac{1}{5}\right) + \frac{d}{d x} \left(- x\right) \]
productApply the product rule to the second term.✓ Proved
\[ = \left(x + \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} \left(x + \frac{1}{5}\right) - 1 \]
constantDifferentiate the first 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 algebraDifferentiate the first part of the product term. Simplify the constant derivative.✓ Proved
\[ = \frac{\left(x + \frac{1}{5}\right) \frac{d}{d x} \left(5 x + 1\right)}{5 x + 1} + \ln{\left(5 x + 1 \right)} - 1 \]
chainApply the chain rule to 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 of the logarithm.✓ Proved
\[ = \ln{\left(5 x + 1 \right)} \]
algebra algebra algebraMultiply the constant 5 into the fraction. Simplify the fraction. Combine the constant terms.✓ 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 undefined where 5*x + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0
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
11	✓ 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: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14
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.