Home›Calculus 1›Product rule›Problem 2.2262

Derivative of \( \displaystyle - x^{2} + \frac{x}{2} + \left(2 x^{2} - x + \frac{1}{8}\right) \ln{\left(4 x - 1 \right)} \)

Problem 2.2262 · hard Video

Differentiate \( \displaystyle f(x) = - x^{2} + \frac{x}{2} + \left(2 x^{2} - x + \frac{1}{8}\right) \ln{\left(4 x - 1 \right)} \).

\[ \frac{d}{d x} \left(- x^{2} + \frac{x}{2} + \left(2 x^{2} - x + \frac{1}{8}\right) \ln{\left(4 x - 1 \right)}\right) \]
sumStart with the derivative of the entire function.✓ Proved
\[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} \left(2 x^{2} - x + \frac{1}{8}\right) \ln{\left(4 x - 1 \right)} \]
sumApply the sum rule to separate the terms.✓ Proved
\[ = \left(2 x^{2} - x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} \left(2 x^{2} - x + \frac{1}{8}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) \]
productApply the product rule to the third term.✓ Proved
\[ = \left(2 x^{2} - x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \left(\frac{d}{d x} \frac{1}{8} - \frac{d}{d x} x + \frac{d}{d x} 2 x^{2}\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) \]
sumApply the sum rule to the polynomial part.✓ Proved
\[ = \left(- \frac{d}{d x} x + \frac{d}{d x} 2 x^{2}\right) \ln{\left(4 x - 1 \right)} + \left(2 x^{2} - x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) \]
constantThe derivative of the constant 1/8 is 0.✓ Proved
\[ = \left(\frac{d}{d x} 2 x^{2} - 1\right) \ln{\left(4 x - 1 \right)} + \left(2 x^{2} - x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) \]
derivativeThe derivative of x is 1.✓ Proved
\[ = \left(4 x - 1\right) \ln{\left(4 x - 1 \right)} + \left(2 x^{2} - x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) \]
derivativeThe derivative of 2*x**2 is 4*x.✓ Proved
\[ = \left(4 x - 1\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) + \frac{\left(2 x^{2} - x + \frac{1}{8}\right) \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \left(4 x - 1\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) + \frac{4 \left(2 x^{2} - x + \frac{1}{8}\right)}{4 x - 1} \]
derivativeThe derivative of 4*x - 1 is 4.✓ Proved
\[ = \left(4 x - 1\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- x^{2}\right) + \frac{8 x^{2} - 4 x + \frac{1}{2}}{4 x - 1} \]
algebraMultiply the 4 into the numerator.✓ Proved
\[ = - 2 x + \left(4 x - 1\right) \ln{\left(4 x - 1 \right)} + \frac{1}{2} + \frac{8 x^{2} - 4 x + \frac{1}{2}}{4 x - 1} \]
derivativeEvaluate the derivatives of the remaining simple terms.✓ Proved
\[ = - 2 x + \left(4 x - 1\right) \ln{\left(4 x - 1 \right)} + \frac{1}{2} + \frac{16 x^{2} - 8 x + 1}{8 x - 2} \]
algebra simplifySimplify the fraction by multiplying numerator and denominator by 2. Final simplification of the expression.✓ Proved

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

Top: the graph of f = -x² + x/2 + (2x² - x + 1/8)·log(4x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2x + (4x - 1)·log(4x - 1) + 1/2 + 4(2x² - x + 1/8)/(4x - 1).

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. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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 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 undefined where 4*x - 1 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4x - 1 = 0 undefined where 8x - 2 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 8*x - 2 = 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 11 applies the 'derivative' rule to evaluate Derivative(-x**2, x) and Derivative(x/2, x), but it fails to evaluate the remaining term (8*x**2 - 4*x + 1/2)/(4*x - 1). The 'derivative' rule is for unfolding d/dx on known forms, not for partial evaluation of a sum. Furthermore, the resulting expression in Step 11 is not equal to the stated answer, and subsequent steps do not simplify it to the stated answer, making the solution mathematically incorrect relative to the task.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — Step 11 applies the 'derivative' rule to two separate terms (-x**2 and x/2) simultaneously, violating the constraint that each step must change only one thing. Additionally, the final result in Step 13 does not match the stated answer, as the rational term (16*x**2 - 8*x + 1)/(8*x - 2) simplifies to 2*x - 1/2, which cancels the linear terms, leaving only (4*x - 1)*log(4*x - 1), but the solution fails to perform this cancellation.
gpt-oss:20b: fail (error) 2026-10-14 — The final expression contains extra terms (-2*x + 1/2 + (16*x**2 - 8*x + 1)/(8*x - 2)) that do not cancel, so the derivative is not simplified to the correct result (4*x - 1)*log(4*x - 1).

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 2026-10-17 with SymPy 1.14.0.