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

Problem 2.44 · medium

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

\[ \frac{d}{d x} \frac{5 \ln{\left(2 x + 2 \right)}}{2} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{5 \frac{d}{d x} \ln{\left(2 x + 2 \right)}}{2} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(2 x + 2\right)}{2 \left(2 x + 2\right)} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \frac{5}{2 x + 2} \]
derivative simplifyDifferentiate the inner function. Simplify the expression by canceling the 2.✓ Proved

Answer \( \frac{5}{2 x + 2} \)

Mind the domain. The answer is also defined on (-oo, -1), where f(x) is not. Substituting there gives a number that is not a slope of f.

✓ 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.

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 undefined where 2*x + 2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in distinct steps. The final simplification is algebraically correct and properly labeled.

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in distinct steps. The final simplification is algebraically correct and properly labeled.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The labels are accurate and the logic is sound.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in distinct steps. The final simplification is algebraically correct.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in separate steps. The final simplification is algebraically correct and properly labeled.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: fail 2026-09-17 — The solution omits the domain restriction x>-1 required for the logarithm, so the simplified derivative 5/(2*x+2) is only valid on that domain.
deepseek-r1:70b: pass 2026-09-17

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 2026-09-26 with SymPy 1.14.0.