Home›Calculus 1›Derivatives of logarithms›Problem 2.2393

Derivative of \( \displaystyle - 2 x + \left(2 x - \frac{2}{5}\right) \ln{\left(5 x - 1 \right)} \)

Problem 2.2393 · hard Video

Differentiate \( \displaystyle f(x) = - 2 x + \left(2 x - \frac{2}{5}\right) \ln{\left(5 x - 1 \right)} \).

\[ \frac{d}{d x} \left(- 2 x + \left(2 x - \frac{2}{5}\right) \ln{\left(5 x - 1 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(2 x - \frac{2}{5}\right) \ln{\left(5 x - 1 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \left(2 x - \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \ln{\left(5 x - 1 \right)} \frac{d}{d x} \left(2 x - \frac{2}{5}\right) + \frac{d}{d x} \left(- 2 x\right) \]
productApply the product rule to the second term.✓ Proved
\[ = \left(2 x - \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \left(- \frac{d}{d x} \frac{2}{5} + \frac{d}{d x} 2 x\right) \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \left(- 2 x\right) \]
sumApply the sum rule to the first part of the product.✓ Proved
\[ = \left(2 x - \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \ln{\left(5 x - 1 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \left(- 2 x\right) \]
constant simplifyThe derivative of the constant 2/5 is 0. Simplify the expression.✓ Proved
\[ = \left(2 x - \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \ln{\left(5 x - 1 \right)} \frac{d}{d x} 2 x - 2 \]
derivativeDifferentiate the linear term.✓ Proved
\[ = \left(2 x - \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + 2 \ln{\left(5 x - 1 \right)} - 2 \]
derivativeDifferentiate the term 2*x.✓ Proved
\[ = \frac{\left(2 x - \frac{2}{5}\right) \frac{d}{d x} \left(5 x - 1\right)}{5 x - 1} + 2 \ln{\left(5 x - 1 \right)} - 2 \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \frac{5 \left(2 x - \frac{2}{5}\right)}{5 x - 1} + 2 \ln{\left(5 x - 1 \right)} - 2 \]
derivativeDifferentiate the inner function 5*x - 1.✓ Proved
\[ = 2 \ln{\left(5 x - 1 \right)} - 2 + \frac{10 x - 2}{5 x - 1} \]
algebra algebraMultiply the numerator by 5. Factor out a 2 from the numerator.✓ Proved
\[ = 2 \ln{\left(5 x - 1 \right)} \]
simplify simplifyCancel the common term (5*x - 1). Combine the remaining constant terms.✓ Proved

Answer \( 2 \ln{\left(5 x - 1 \right)} \)

Top: the graph of f = -2x + (2x - 2/5)·log(5x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5(2x - 2/5)/(5x - 1) + 2log(5x - 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 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 undefined where 5*x - 1 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0
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: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. All labels are appropriate for the transformations performed.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint. The labels used are appropriate for the operations performed.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.