Home›Calculus 1›Product rule›Problem 2.2555

Derivative of \( \displaystyle - \frac{5 x^{2}}{4} + \frac{x}{2} + \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) \ln{\left(5 x - 1 \right)} \)

Problem 2.2555 · hard Video

Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{4} + \frac{x}{2} + \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) \ln{\left(5 x - 1 \right)} \).

\[ \frac{d}{d x} \left(- \frac{5 x^{2}}{4} + \frac{x}{2} + \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) \ln{\left(5 x - 1 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) \ln{\left(5 x - 1 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \ln{\left(5 x - 1 \right)} \frac{d}{d x} \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) \]
productApply the product rule to the third term.✓ Proved
\[ = \left(5 x - 1\right) \ln{\left(5 x - 1 \right)} + \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) \]
derivativeDifferentiate the first part of the product.✓ Proved
\[ = \left(5 x - 1\right) \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) + \frac{\left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) \frac{d}{d x} \left(5 x - 1\right)}{5 x - 1} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \left(5 x - 1\right) \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) + \frac{5 \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right)}{5 x - 1} \]
derivativeDifferentiate the inner function of the chain rule.✓ Proved
\[ = \left(5 x - 1\right) \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) + \frac{\frac{25 x^{2}}{2} - 5 x + \frac{1}{2}}{5 x - 1} \]
algebraSimplify the product of the fraction and 5.✓ Proved
\[ = - \frac{5 x}{2} + \left(5 x - 1\right) \ln{\left(5 x - 1 \right)} + \frac{1}{2} + \frac{\frac{25 x^{2}}{2} - 5 x + \frac{1}{2}}{5 x - 1} \]
derivative algebra simplifyEvaluate the remaining derivatives. Prepare for simplification. Combine the terms into the final simplified form.✓ Proved

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

Top: the graph of f = -5x²/4 + x/2 + (5x²/2 - x + 1/10)·log(5x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -5x/2 + (5x - 1)·log(5x - 1) + 1/2 + 5(5x²/2 - x + 1/10)/(5x - 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 undefined where 5*x - 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0
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 undefined where 5*x - 1 = 0
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. The algebraic simplification in steps 9 and 10, while verbose (adding and subtracting the same term), is mathematically valid and correctly identifies that the rational part simplifies to cancel the polynomial terms, leading to the correct final answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-10 — The solution correctly applies differentiation rules step-by-step. The algebraic simplification in steps 9 and 10, while verbose (adding and subtracting the same term), is mathematically valid and correctly identifies that the rational part simplifies to cancel the polynomial terms, leading to the correct final answer.
gpt-oss:20b: pass 2026-11-10
qwen3.6:27b-mlx: fail (style) 2026-11-09 — Steps 9 and 10 are redundant and do not contribute to the derivation; Step 9 adds and subtracts a term, and Step 10 cancels them, which is unnecessary clutter. Additionally, Step 7 performs a polynomial division that simplifies the fraction to a linear term plus a remainder, but the solution leaves it in unsimplified form, making the subsequent cancellation in Step 9 look like a trick rather than a natural simplification.
gpt-oss:20b: fail (style) 2026-11-09 — Step 9 applies two algebraic operations (adding a term and subtracting the same term) in a single step, violating the rule that each step must change only one thing.
qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly applies differentiation rules step-by-step. The algebraic simplification in steps 9 and 10, while verbose, is valid and leads to the correct result.
gpt-oss:20b: pass 2026-11-09

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 2026-11-10 with SymPy 1.14.0.