Home›Calculus 1›Product rule›Problem 2.2283

Derivative of \( \displaystyle - 2 x^{2} + 3 x + \left(4 x^{2} - 6 x + \frac{9}{4}\right) \ln{\left(4 x - 3 \right)} \)

Problem 2.2283 · hard Video

Differentiate \( \displaystyle f(x) = - 2 x^{2} + 3 x + \left(4 x^{2} - 6 x + \frac{9}{4}\right) \ln{\left(4 x - 3 \right)} \).

\[ \frac{d}{d x} \left(- 2 x^{2} + 3 x + \left(4 x^{2} - 6 x + \frac{9}{4}\right) \ln{\left(4 x - 3 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{d}{d x} \left(4 x^{2} - 6 x + \frac{9}{4}\right) \ln{\left(4 x - 3 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \left(4 x^{2} - 6 x + \frac{9}{4}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} \left(4 x^{2} - 6 x + \frac{9}{4}\right) + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 2 x^{2}\right) \]
productApply the product rule to the third term.✓ Proved
\[ = \left(8 x - 6\right) \ln{\left(4 x - 3 \right)} + \left(4 x^{2} - 6 x + \frac{9}{4}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 2 x^{2}\right) \]
derivativeDifferentiate the polynomial part of the product.✓ Proved
\[ = \left(8 x - 6\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{\left(4 x^{2} - 6 x + \frac{9}{4}\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \left(8 x - 6\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{4 \left(4 x^{2} - 6 x + \frac{9}{4}\right)}{4 x - 3} \]
derivativeDifferentiate the inner function of the chain rule.✓ Proved
\[ = \left(8 x - 6\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{16 x^{2} - 24 x + 9}{4 x - 3} \]
algebraMultiply the numerator by 4.✓ Proved
\[ = 4 x + \left(8 x - 6\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 2 x^{2}\right) - 3 \]
algebra simplifyRecognize the numerator as a perfect square trinomial. Simplify the fraction.✓ Proved
\[ = \left(8 x - 6\right) \ln{\left(4 x - 3 \right)} \]
derivative simplifyEvaluate the remaining derivatives. Combine like terms to get the final result.✓ Proved

Answer \( 2 \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} \)

Top: the graph of f = -2x² + 3x + (4x² - 6x + 9/4)·log(4x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -4x + (8x - 6)·log(4x - 3) + 3 + 4(4x² - 6x + 9/4)/(4x - 3).

✓ 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 4*x - 3 = 0
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
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, simplifying the resulting expression to match the stated answer. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies differentiation rules step-by-step, simplifying the resulting expression to match the stated answer. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and final result are correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and rule applications are accurate.
gpt-oss:20b: pass 2026-10-14

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.