Home›Calculus 1›Derivatives of logarithms›Problem 2.2404

Derivative of \( \displaystyle - 3 x + \left(3 x + \frac{3}{2}\right) \ln{\left(2 x + 1 \right)} \)

Problem 2.2404 · hard Video

Differentiate \( \displaystyle f(x) = - 3 x + \left(3 x + \frac{3}{2}\right) \ln{\left(2 x + 1 \right)} \).

\[ \frac{d}{d x} \left(- 3 x + \left(3 x + \frac{3}{2}\right) \ln{\left(2 x + 1 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x + \frac{3}{2}\right) \ln{\left(2 x + 1 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x \ln{\left(2 x + 1 \right)} + \frac{3 \ln{\left(2 x + 1 \right)}}{2}\right) \]
algebraDistribute the term.✓ Proved
\[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} 3 x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} \]
sumApply the sum rule again.✓ Proved
\[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} 3 x \ln{\left(2 x + 1 \right)} + \frac{3 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} \]
constant-multipleFactor out the constant 3/2.✓ Proved
\[ = 3 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 3 x\right) + \frac{3 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} \]
productApply the product rule to the middle term.✓ Proved
\[ = 3 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 3 \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} \]
derivativeDifferentiate 3*x.✓ Proved
\[ = 3 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 3 \ln{\left(2 x + 1 \right)} + \frac{3 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 3 \]
constant algebraDifferentiate -3*x. Remove the parentheses.✓ Proved
\[ = \left(3 x + \frac{3}{2}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 3 \ln{\left(2 x + 1 \right)} - 3 \]
algebraGroup the terms with the derivative.✓ Proved
\[ = 3 \ln{\left(2 x + 1 \right)} - 3 + \frac{\left(3 x + \frac{3}{2}\right) \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = 3 \ln{\left(2 x + 1 \right)} - 3 + \frac{2 \left(3 x + \frac{3}{2}\right)}{2 x + 1} \]
derivativeDifferentiate 2*x + 1.✓ Proved
\[ = 3 \ln{\left(2 x + 1 \right)} - 3 + \frac{6 x + 3}{2 x + 1} \]
algebra algebraMultiply the numerator. Factor out 3 from the numerator.✓ Proved
\[ = 3 \ln{\left(2 x + 1 \right)} \]
simplify simplifyCancel the common factor (2*x + 1). Combine the remaining constants.✓ Proved

Answer \( 3 \ln{\left(2 x + 1 \right)} \)

Top: the graph of f = -3x + (3x + 3/2)·log(2x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3log(2x + 1) - 3 + 2(3x + 3/2)/(2x + 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
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
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 undefined where 2*x + 1 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
15	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: pass 2026-10-16
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 and using valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.