Derivative of \( \displaystyle \ln{\left(\left(4 x + 1\right)^{2} + 1 \right)} \)
Problem 2.634 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\left(4 x + 1\right)^{2} + 1 \right)} \).
- \[ \frac{d}{d x} \ln{\left(\left(4 x + 1\right)^{2} + 1 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(4 x + 1\right)^{2} + 1\right)}{\left(4 x + 1\right)^{2} + 1} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(4 x + 1\right)^{2}}{\left(4 x + 1\right)^{2} + 1} \]sumApply the sum rule to the inner expression.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(4 x + 1\right)^{2}}{\left(4 x + 1\right)^{2} + 1} \]constant algebraThe derivative of the constant 1 is 0. Simplify the expression.✓ Proved
- \[ = \frac{\left(8 x + 2\right) \frac{d}{d x} \left(4 x + 1\right)}{\left(4 x + 1\right)^{2} + 1} \]powerApply the power rule and chain rule to the squared term.✓ Proved
- \[ = \frac{32 x + 8}{\left(4 x + 1\right)^{2} + 1} \]derivative algebraDifferentiate the linear term 4x + 1. Multiply the constants together.✓ Proved
- \[ = \frac{32 x + 8}{16 x^{2} + 8 x + 2} \]algebra simplifyExpand the denominator. Final simplified form.✓ Proved
Answer \( \frac{32 x + 8}{\left(4 x + 1\right)^{2} + 1} \)
✓ 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.
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 (4*x + 1)**2 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 undefined where 16*x**2 + 8*x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**2 + 8*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (4*x + 1)**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: passqwen3.6:27b-mlx: fail (style) — Step 6 applies both the power rule and the chain rule simultaneously, violating the contract that each step must change only one thing. It should be split into a step applying the power rule to get 2*(4*x+1)*Derivative(4*x+1, x) is incorrect because the power rule alone on (u)^2 gives 2*u*u', so it is actually applying the generalized power rule which combines power and chain. A strict single-rule step would apply the chain rule to the outer square function first or use a specific 'generalized power' label if available, but since 'power' usually implies d/dx(x^n), applying it to a composite function without an explicit chain step is a granularity defect. Specifically, Step 6 claims rule 'power' but performs the differentiation of the composite function (4x+1)^2 entirely, which requires the chain rule. The label 'power' is insufficient/misleading for a composite function differentiation step that doesn't explicitly invoke 'chain' in that step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 6 applies both the power rule and the chain rule simultaneously, violating the contract that each step must change only one thing. It should be split into a step applying the power rule to get 2*(4*x+1)*Derivative(4*x+1, x) is incorrect because the power rule alone on (u)^2 gives 2*u*u', so it is actually applying the generalized power rule which combines power and chain. A strict single-rule step would apply the chain rule to the outer square function first or use a specific 'generalized power' label if available, but since 'power' usually implies d/dx(x^n), applying it to a composite function without an explicit chain step is a granularity defect. Specifically, Step 6 claims rule 'power' but performs the differentiation of the composite function (4x+1)^2 entirely, which requires the chain rule. The label 'power' is insufficient/misleading for a composite function differentiation step that doesn't explicitly invoke 'chain' in that step.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 6 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. It should be split into a power rule step followed by a chain rule step.gpt-oss:20b: pass 2026-09-21
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.