Home›Calculus 1›Product rule›Problem 2.2461

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

Problem 2.2461 · hard Video

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

\[ \frac{d}{d x} \left(- \frac{9 x^{2}}{4} - 3 x + \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{9 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} \]
sumApply the sum rule.✓ Proved
\[ = - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} \]
constant-multiplePull out the constant -9/4.✓ Proved
\[ = \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} \]
productApply the product rule to the third term.✓ Proved
\[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} + \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} \]
derivative algebraDifferentiate the polynomial inside the product. Simplify the polynomial derivative.✓ Proved
\[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{\left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{3 \left(\frac{9 x^{2}}{2} + 6 x + 2\right)}{3 x + 2} \]
derivativeDifferentiate the inner function 3*x + 2.✓ Proved
\[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{\frac{27 x^{2}}{2} + 18 x + 6}{3 x + 2} \]
algebraDistribute the 3 into the numerator.✓ Proved
\[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{27 x^{2} + 36 x + 12}{6 x + 4} \]
algebra algebraFind a common denominator for the fraction term. Factor out 3 from the numerator.✓ Proved
\[ = \frac{3 \left(3 x + 2\right)^{2}}{6 x + 4} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} \]
algebraRecognize the perfect square trinomial.✓ Proved
\[ = \frac{9 x}{2} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + 3 \]
algebraCancel the common factor (3*x + 2).✓ Proved
\[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} \]
derivative algebra simplifyEvaluate the remaining derivatives. Distribute and expand the terms. Combine like terms to get the final result.✓ Proved

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

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

✓ 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
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 undefined where 3*x + 2 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3x + 2 = 0 undefined where 6x + 4 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 6*x + 4 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 6*x + 4 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 6*x + 4 = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
15	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
16	✓ 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 and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (style) 2026-10-19 — Step 3 applies the constant-multiple rule to two separate terms simultaneously, violating the one-rule-per-step constraint. Step 14 applies the derivative rule to two separate terms simultaneously, also violating the constraint.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-19 — Step 3 applies the constant-multiple rule to two terms simultaneously (-9/4 and -3), violating the contract that each step must change only one thing. Step 14 also applies the derivative rule to two terms simultaneously.
gpt-oss:20b: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.