Home›Calculus 1›Derivatives of logarithms›Problem 2.1173

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

Problem 2.1173 · hard Video

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

\[ \frac{d}{d x} \left(2 x \ln{\left(2 x + 1 \right)} - 2 x + \ln{\left(2 x + 1 \right)}\right) \]
Differentiate the function with respect to x.✓ Proved
\[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \ln{\left(2 x + 1 \right)} \]
sumApply the sum rule for differentiation.✓ Proved
\[ = \frac{d}{d x} 2 x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \ln{\left(2 x + 1 \right)} - 2 \]
constantThe derivative of 2*x is 2.✓ Proved
\[ = 2 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \ln{\left(2 x + 1 \right)} - 2 \]
productApply the product rule to the first term.✓ Proved
\[ = 2 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 2 \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \ln{\left(2 x + 1 \right)} - 2 \]
constant-multipleThe derivative of 2*x is 2.✓ Proved
\[ = \frac{4 x}{2 x + 1} + 2 \ln{\left(2 x + 1 \right)} - 2 + \frac{2}{2 x + 1} \]
logarithmic algebra algebraApply the chain rule to the log terms. Simplify the derivatives of the log terms. Multiply the terms.✓ Proved
\[ = 2 \ln{\left(2 x + 1 \right)} - 2 + \frac{4 x + 2}{2 x + 1} \]
algebra algebraCombine the fractions. Factor the numerator.✓ Proved
\[ = 2 \ln{\left(2 x + 1 \right)} \]
simplify simplifyCancel the common factor in the fraction. Simplify the final expression.✓ Proved

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

Top: the graph of f = 2x·log(2x + 1) - 2x + log(2x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4x/(2x + 1) + 2log(2x + 1) - 2 + 2/(2x + 1).

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 undefined where 2*x + 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
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
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: fail (error) — Step 3 is labeled 'constant', but it differentiates 2*x (which is not a constant) to get 2. The correct label for differentiating a linear term like 2*x is 'derivative' (or 'constant-multiple' if viewed as 2*d/dx(x), but 'constant' is strictly for d/dx(c)=0). Additionally, Step 6 applies the chain rule inside the logarithmic differentiation, but the label is 'logarithmic'; while 'logarithmic' is acceptable for the outer log derivative, the step explicitly shows the chain rule application (multiplying by derivative of inner function). However, the primary defect is Step 3's incorrect label 'constant' for a non-constant term.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is labeled 'constant', but it differentiates 2*x (which is not a constant) to get 2. The correct label for differentiating a linear term like 2*x is 'derivative' (or 'constant-multiple' if viewed as 2*d/dx(x), but 'constant' is strictly for d/dx(c)=0). Additionally, Step 6 applies the chain rule inside the logarithmic differentiation, but the label is 'logarithmic'; while 'logarithmic' is acceptable for the outer log derivative, the step explicitly shows the chain rule application (multiplying by derivative of inner function). However, the primary defect is Step 3's incorrect label 'constant' for a non-constant term.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29

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 2026-10-03 with SymPy 1.14.0.