Home›Calculus 1›Derivatives of logarithms›Problem 2.2419

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

Problem 2.2419 · hard Video

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

\[ \frac{d}{d x} \left(x - \left(x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} x - \frac{d}{d x} \left(x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)} \]
sumApply the difference rule.✓ Proved
\[ = - \left(x + \frac{1}{3}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} - \ln{\left(3 x + 1 \right)} \frac{d}{d x} \left(x + \frac{1}{3}\right) + \frac{d}{d x} x \]
productApply the product rule to the second term.✓ Proved
\[ = - \left(x + \frac{1}{3}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} - \ln{\left(3 x + 1 \right)} \frac{d}{d x} \frac{1}{3} - \ln{\left(3 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
sumApply the sum rule to the first term of the product.✓ Proved
\[ = - \left(x + \frac{1}{3}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} - \ln{\left(3 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
constant simplifyThe derivative of the constant 1/3 is 0. Simplify the expression by removing the zero term.✓ Proved
\[ = - \frac{\left(x + \frac{1}{3}\right) \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} - \ln{\left(3 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = - \frac{3 \left(x + \frac{1}{3}\right)}{3 x + 1} - \ln{\left(3 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
chainApply the chain rule to the inner function 3x + 1.✓ Proved
\[ = - \frac{3 \left(x + \frac{1}{3}\right)}{3 x + 1} - \ln{\left(3 x + 1 \right)} + 1 \]
derivativeEvaluate the derivatives of the basic terms.✓ Proved
\[ = - \ln{\left(3 x + 1 \right)} \]
algebra algebra algebra simplifyDistribute the negative sign. Distribute the 3 in the numerator. Simplify the fraction. Combine the constant terms.✓ Proved

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

Top: the graph of f = x - (x + 1/3)·log(3x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3(x + 1/3)/(3x + 1) - log(3x + 1) + 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 undefined where 3*x + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
13	✓ 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 2026-10-19
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-16 — Step 4 applies the sum rule to the term (x + 1/3) inside the product, but the label 'sum' is ambiguous because the outer structure is still a product; more critically, Step 8 is labeled 'chain' but performs the evaluation of the derivative of the inner function (a 'derivative' step) and simplification, violating the one-rule-per-step constraint by combining chain rule application with basic differentiation and algebraic simplification.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-16 — Step 8 is labeled 'chain', but it evaluates the derivative of the inner function (a basic power/constant rule), which should be labeled 'derivative'. Step 7 correctly labeled the chain rule application to the log structure, so Step 8 is redundant or mislabeled.
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.