Home›Calculus 1›Chain rule›Problem 2.294

Derivative of \( \displaystyle 5 x \ln{\left(4 x - 3 \right)} - 5 x - \frac{15 \ln{\left(4 x - 3 \right)}}{4} \)

Problem 2.294 · hard

Differentiate \( \displaystyle f(x) = 5 x \ln{\left(4 x - 3 \right)} - 5 x - \frac{15 \ln{\left(4 x - 3 \right)}}{4} \).

\[ \frac{d}{d x} \left(5 x \ln{\left(4 x - 3 \right)} - 5 x - \frac{15 \ln{\left(4 x - 3 \right)}}{4}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{15 \ln{\left(4 x - 3 \right)}}{4} \]
sumApply the sum rule to differentiate each term separately.✓ Proved
\[ = \frac{d}{d x} 5 x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{15 \ln{\left(4 x - 3 \right)}}{4} - 5 \]
constantThe derivative of 5*x is 5.✓ Proved
\[ = 5 x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} 5 x - \frac{d}{d x} \frac{15 \ln{\left(4 x - 3 \right)}}{4} - 5 \]
productApply the product rule to the first term.✓ Proved
\[ = 5 x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + 5 \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{15 \ln{\left(4 x - 3 \right)}}{4} - 5 \]
constant-multipleDifferentiate the first part of the product term.✓ Proved
\[ = \frac{20 x}{4 x - 3} + 5 \ln{\left(4 x - 3 \right)} - 5 - \frac{15}{4 x - 3} \]
chain algebraApply the chain rule to the logarithmic terms. Simplify the products of the constants and the inner derivative.✓ Proved
\[ = 5 \ln{\left(4 x - 3 \right)} - 5 + \frac{20 x - 15}{4 x - 3} \]
algebra algebraCombine the fractions with the same denominator. Factor out 5 from the numerator.✓ Proved
\[ = 5 \ln{\left(4 x - 3 \right)} \]
simplify simplifyCancel the common term (4*x - 3) in the fraction. Combine the remaining constant terms.✓ Proved

Answer \( 5 \log{\left(4 x - 3 \right)} \)

✓ 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.

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 - 3 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
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: fail (style) — Step 5 incorrectly labels the simplification of Derivative(5*x, x) as "constant-multiple"; it should be "constant". No other multi-rule steps are present.
deepseek-r1:70b: 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, and the algebraic simplification is sound.

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20 — 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, and the algebraic simplification is sound.
gpt-oss:20b: fail (style) 2026-09-20 — Step 5 incorrectly labels the simplification of Derivative(5*x, x) as "constant-multiple"; it should be "constant". No other multi-rule steps are present.
qwen3.6:27b-mlx: pass 2026-09-20 — 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 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — 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.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (style) 2026-09-19 — Several steps are labeled with the wrong rule. Step 3 applies the derivative rule to 5*x, not the constant rule. Step 5 simplifies the derivative of 5*x to 5, again a derivative step, not a constant‑multiple step. These mislabelings mislead a student about which rule is being used.
qwen3.6:27b-mlx: pass 2026-09-19 — 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.
deepseek-r1:70b: fail (style) 2026-09-19 — First step lacks a rule label.
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: fail 2026-09-17 — Step 3 incorrectly labels the derivative of 5*x as "constant"; it should be the constant‑multiple rule (derivative of c·x = c).
deepseek-r1:70b: pass 2026-09-17

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.