Home›Calculus 1›Derivatives of logarithms›Problem 2.370

Derivative of \( \displaystyle \ln{\left(16 x^{2} + 1 \right)} \)

Problem 2.370 · hard Video

Differentiate \( \displaystyle f(x) = \ln{\left(16 x^{2} + 1 \right)} \).

\[ \frac{d}{d x} \ln{\left(16 x^{2} + 1 \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \left(16 x^{2} + 1\right)}{16 x^{2} + 1} \]
chainApply the chain rule for the natural logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} 16 x^{2}}{16 x^{2} + 1} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = \frac{32 x}{16 x^{2} + 1} \]
derivative constant algebraDifferentiate each term separately. The derivative of the constant 1 is 0. Multiply the terms together.✓ Proved

Answer \( \frac{32 x}{16 x^{2} + 1} \)

Top: the graph of f = log(16x² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 32x/(16x² + 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 undefined where 16x*2 + 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x*2 + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x*2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x*2 + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x*2 + 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 16x*2 + 1 = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: fail (error) — Step 4 applies the derivative rule to two terms at once (32*x + 0), violating the rule‑by‑step requirement that each step change only one thing.
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-08: pass — Every line is correct and SymPy proved each step. The rule labels are acceptable and no step does two independent things.

gpt-oss:20b: dismiss — Step 4 evaluates the derivatives of 16*x**2 and 1 inside one sum; applying the derivative rule to the terms of that sum is one change, and the 'derivative' label is legitimate. SymPy also proved the line.

Every verdict on record (11)

claude-sonnet-5-5: pass 2026-10-08 — Every line is correct and SymPy proved each step. The rule labels are acceptable and no step does two independent things.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies the derivative rule to two terms at once (32*x + 0), violating the rule‑by‑step requirement that each step change only one thing.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (style) 2026-09-20 — Step 5 removes the zero term using simplification, but labels it as "constant". The correct label for eliminating the zero is "simplify".
qwen3.6:27b-mlx: pass 2026-09-19
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 chain rule, sum rule, and power rule in distinct steps. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19

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.