Derivative of \( \displaystyle \ln{\left(\left(4 x + 1\right)^{2} + 1 \right)} \)
Problem 2.634 · hard Video
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. 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 (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 one-rule-per-step constraint. 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', which is what is shown, but the label 'power' usually implies the basic rule d/dx(x^n)=nx^(n-1). Applying it to a composite function requires the chain rule. The step shows the result of applying the chain rule to the power function. If the rule is 'power', it implies d/dx(u^2) = 2u, missing the u'. If the rule is 'chain', it implies d/dx(f(g)) = f'(g)g'. The step combines them. A better split would be: apply chain rule to (4*x+1)^2 to get 2*(4*x+1)*Derivative(4*x+1, x) labeled 'chain' (or 'power' if the system treats power rule as including the inner derivative, but standard calculus pedagogy separates them). Given the strict 'one thing' rule, applying the derivative of the outer function (power) and the derivative of the inner function (chain) in one step is a defect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. 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', which is what is shown, but the label 'power' usually implies the basic rule d/dx(x^n)=nx^(n-1). Applying it to a composite function requires the chain rule. The step shows the result of applying the chain rule to the power function. If the rule is 'power', it implies d/dx(u^2) = 2u, missing the u'. If the rule is 'chain', it implies d/dx(f(g)) = f'(g)g'. The step combines them. A better split would be: apply chain rule to (4*x+1)^2 to get 2*(4*x+1)*Derivative(4*x+1, x) labeled 'chain' (or 'power' if the system treats power rule as including the inner derivative, but standard calculus pedagogy separates them). Given the strict 'one thing' rule, applying the derivative of the outer function (power) and the derivative of the inner function (chain) in one step is a defect.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-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.