Home›Calculus 1›Derivatives of logarithms›Problem 2.2509

Derivative of \( \displaystyle - 5 x + \left(5 x + \frac{10}{3}\right) \ln{\left(3 x + 2 \right)} \)

Problem 2.2509 · hard Video

Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x + \frac{10}{3}\right) \ln{\left(3 x + 2 \right)} \).

\[ \frac{d}{d x} \left(- 5 x + \left(5 x + \frac{10}{3}\right) \ln{\left(3 x + 2 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x + \frac{10}{3}\right) \ln{\left(3 x + 2 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \frac{10 \ln{\left(3 x + 2 \right)}}{3} \]
productApply the product rule to the second term.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(3 x + 2 \right)} + \frac{10 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} \]
constant-multipleApply the constant multiple rule to the third term.✓ Proved
\[ = \frac{d}{d x} 5 x \ln{\left(3 x + 2 \right)} + \frac{10 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 5 \]
constantDifferentiate the first term.✓ Proved
\[ = 5 x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} 5 x + \frac{10 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 5 \]
productApply the product rule to the second term.✓ Proved
\[ = 5 x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + 5 \ln{\left(3 x + 2 \right)} + \frac{10 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 5 \]
constantDifferentiate the first term of the product.✓ Proved
\[ = \frac{15 x}{3 x + 2} + 5 \ln{\left(3 x + 2 \right)} + \frac{10 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 5 \]
chainApply the chain rule to the log terms.✓ Proved
\[ = \frac{15 x}{3 x + 2} + 5 \ln{\left(3 x + 2 \right)} - 5 + \frac{10}{3 x + 2} \]
chain algebraApply the chain rule to the third term. Simplify the terms.✓ Proved
\[ = 5 \ln{\left(3 x + 2 \right)} - 5 + \frac{15 x + 10}{3 x + 2} \]
algebra algebraCombine the fractions. Factor the numerator.✓ Proved
\[ = 5 \ln{\left(3 x + 2 \right)} \]
simplify simplifyCancel the common factor. Combine the constant terms.✓ Proved

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

Top: the graph of f = -5x + (5x + 10/3)·log(3x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5log(3x + 2) - 5 + 3(5x + 10/3)/(3x + 2).

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 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 3*x + 2 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0
14	✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-27
gpt-oss:20b: pass 2026-10-27
qwen3.6:27b-mlx: pass 2026-10-26 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications are valid and clearly labeled.
gpt-oss:20b: fail (error) 2026-10-26 — Step 3 incorrectly applies the product rule to the term (5*x + 10/3)*log(3*x + 2). The correct approach is to first rewrite it as 5*x*log(3*x + 2) + (10/3)*log(3*x + 2) (an algebraic split) and then apply the product rule only to the 5*x*log part. The step as written conflates two separate operations into one, violating the single‑rule‑per‑step rule.
qwen3.6:27b-mlx: fail (style) 2026-10-26 — Step 3 is labeled 'product' but performs a distribution (algebraic expansion) of the term (5*x + 10/3)*log(...) into a sum, not the product rule differentiation. The product rule is correctly applied in Step 6.
gpt-oss:20b: fail (style) 2026-10-26 — Step 3 incorrectly labels a distribution (sum) as a product. The step applies the sum rule, not the product rule, so the rule name is misleading.

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 2026-10-27 with SymPy 1.14.0.