Home›Calculus 1›Product rule›Problem 2.1930

Derivative of \( \displaystyle - \frac{5 x^{2}}{2} + 3 x + \left(5 x^{2} - 6 x + \frac{9}{5}\right) \ln{\left(5 x - 3 \right)} \)

Problem 2.1930 · hard

Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{2} + 3 x + \left(5 x^{2} - 6 x + \frac{9}{5}\right) \ln{\left(5 x - 3 \right)} \).

\[ \frac{d}{d x} \left(- \frac{5 x^{2}}{2} + 3 x + \left(5 x^{2} - 6 x + \frac{9}{5}\right) \ln{\left(5 x - 3 \right)}\right) \]
sumStart with the derivative of the entire function.✓ Proved
\[ = \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) + \frac{d}{d x} \left(5 x^{2} - 6 x + \frac{9}{5}\right) \ln{\left(5 x - 3 \right)} \]
sumApply the sum rule to separate the terms.✓ Proved
\[ = \left(5 x^{2} - 6 x + \frac{9}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x^{2} - 6 x + \frac{9}{5}\right) + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) \]
productApply the product rule to the third term.✓ Proved
\[ = \left(10 x - 6\right) \ln{\left(5 x - 3 \right)} + \left(5 x^{2} - 6 x + \frac{9}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) \]
derivative algebraDifferentiate the polynomial part of the product. Simplify the polynomial expression.✓ Proved
\[ = \left(10 x - 6\right) \ln{\left(5 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) + \frac{\left(5 x^{2} - 6 x + \frac{9}{5}\right) \frac{d}{d x} \left(5 x - 3\right)}{5 x - 3} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \left(10 x - 6\right) \ln{\left(5 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) + \frac{5 \left(5 x^{2} - 6 x + \frac{9}{5}\right)}{5 x - 3} \]
derivative algebraDifferentiate the inner function of the logarithm. Simplify the expression.✓ Proved
\[ = \left(10 x - 6\right) \ln{\left(5 x - 3 \right)} + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) + 3 + \frac{5 \left(5 x^{2} - 6 x + \frac{9}{5}\right)}{5 x - 3} \]
derivativeDifferentiate the first two terms.✓ Proved
\[ = - 5 x + \left(10 x - 6\right) \ln{\left(5 x - 3 \right)} + 3 + \frac{5 \left(5 x^{2} - 6 x + \frac{9}{5}\right)}{5 x - 3} \]
derivativeDifferentiate the remaining power term.✓ Proved
\[ = - 5 x + \left(10 x - 6\right) \ln{\left(5 x - 3 \right)} + 3 + \frac{25 x^{2} - 30 x + 9}{5 x - 3} \]
algebraDistribute the 5 in the last term.✓ Proved
\[ = \left(10 x - 6\right) \ln{\left(5 x - 3 \right)} \]
algebra simplifyRecognize the numerator as a perfect square. Simplify the fraction and combine terms.✓ Proved

Answer \( 2 \left(5 x - 3\right) \ln{\left(5 x - 3 \right)} \)

✓ 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 5*x - 3 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0
13	✓ 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 9 applies the derivative rule to two separate terms simultaneously, violating the constraint that each step must change only one thing. Step 10 then differentiates the remaining term, meaning the differentiation of the first two terms was split across two steps but performed in a single logical jump in step 9.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (style) 2026-10-09 — Step 9 applies the derivative rule to two separate terms simultaneously, violating the constraint that each step must change only one thing. Step 10 then differentiates the remaining term, meaning the differentiation of the first two terms was split across two steps but performed in a single logical jump in step 9.
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications are valid and lead to the correct final result.
gpt-oss:20b: pass 2026-10-09

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 2026-10-09 with SymPy 1.14.0.