Home›Calculus 1›Derivatives of logarithms›Problem 2.2423

Derivative of \( \displaystyle x + \left(\frac{1}{4} - x\right) \ln{\left(4 x - 1 \right)} \)

Problem 2.2423 · hard Video

Differentiate \( \displaystyle f(x) = x + \left(\frac{1}{4} - x\right) \ln{\left(4 x - 1 \right)} \).

\[ \frac{d}{d x} \left(x + \left(\frac{1}{4} - x\right) \ln{\left(4 x - 1 \right)}\right) \]
Start with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} x + \frac{d}{d x} \left(\frac{1}{4} - x\right) \ln{\left(4 x - 1 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \frac{d}{d x} \left(\frac{1}{4} - x\right) \ln{\left(4 x - 1 \right)} + 1 \]
constantDifferentiate the first term.✓ Proved
\[ = \left(\frac{1}{4} - x\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} \left(\frac{1}{4} - x\right) + 1 \]
productApply the product rule to the second term.✓ Proved
\[ = \left(\frac{1}{4} - x\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} - \ln{\left(4 x - 1 \right)} + 1 \]
derivativeDifferentiate the first part of the product.✓ Proved
\[ = \frac{\left(\frac{1}{4} - x\right) \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} - \ln{\left(4 x - 1 \right)} + 1 \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \frac{4 \left(\frac{1}{4} - x\right)}{4 x - 1} - \ln{\left(4 x - 1 \right)} + 1 \]
derivativeDifferentiate the inner function of the logarithm.✓ Proved
\[ = \frac{1 - 4 x}{4 x - 1} - \ln{\left(4 x - 1 \right)} + 1 \]
algebra algebraSimplify the expression. Distribute the 4.✓ Proved
\[ = - \ln{\left(4 x - 1 \right)} \]
algebra simplify simplifyFactor out a negative sign from the numerator. Simplify the fraction. Combine like terms.✓ Proved

Answer \( - \ln{\left(4 x - 1 \right)} \)

Top: the graph of f = x + (1/4 - x)·log(4x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4(1/4 - x)/(4x - 1) - log(4x - 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 undefined where 4*x - 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
12	✓ 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 in a step-by-step manner, 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 2026-10-19 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
gpt-oss:20b: pass 2026-10-19
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 and the final result is correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint. All labels are appropriate for the operations performed.
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.