Derivative of \( \displaystyle \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \)
Problem 2.597 · hard
Differentiate \( \displaystyle f(x) = \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \).
- \[ \frac{d}{d x} \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\ln{\left(4 x + 1 \right)} \sin{\left(4 x + 1 \right)}} \]rewriteRewrite the power using the exponential and logarithm.≈ Checked numerically
- \[ = e^{\ln{\left(4 x + 1 \right)} \sin{\left(4 x + 1 \right)}} \frac{d}{d x} \ln{\left(4 x + 1 \right)} \sin{\left(4 x + 1 \right)} \]chainApply the chain rule to the exponential function.✓ Proved
- \[ = \left(\ln{\left(4 x + 1 \right)} \frac{d}{d x} \sin{\left(4 x + 1 \right)} + \sin{\left(4 x + 1 \right)} \frac{d}{d x} \ln{\left(4 x + 1 \right)}\right) e^{\ln{\left(4 x + 1 \right)} \sin{\left(4 x + 1 \right)}} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(4 \ln{\left(4 x + 1 \right)} \cos{\left(4 x + 1 \right)} + \frac{4 \sin{\left(4 x + 1 \right)}}{4 x + 1}\right) e^{\ln{\left(4 x + 1 \right)} \sin{\left(4 x + 1 \right)}} \]chain algebraApply the chain rule to the sine and logarithm terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \left(4 \ln{\left(4 x + 1 \right)} \cos{\left(4 x + 1 \right)} + \frac{4 \sin{\left(4 x + 1 \right)}}{4 x + 1}\right) \]simplifySubstitute back the original base and exponent.≈ Checked numerically
Answer \( \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \left(4 \log{\left(4 x + 1 \right)} \cos{\left(4 x + 1 \right)} + \frac{4 \sin{\left(4 x + 1 \right)}}{4 x + 1}\right) \)
Lines: 6 proved, 2 checked numerically. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*((4*x + 1)*log(4*x + 1)*cos(4*x + 1) + sin(4*x + 1))*((4*x + 1)**sin(4*x + 1) - exp(log(4*x + 1)*sin(4*x + 1)))/(4*x + 1); numeric agreement only, at 24 of 24 sampled points 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 undefined where 4*x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*((4*x + 1)*log(4*x + 1)*cos(4*x + 1) + sin(4*x + 1))*(-(4*x + 1)**sin(4*x + 1) + exp(log(4*x + 1)*sin(4*x + 1)))/(4*x + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 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 undefined where 4*x + 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 5 applies the chain rule twice (to the sine and to the logarithm) in a single line, violating the rule that each step must change only one thing. The label "chain" is therefore incorrect for that step.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies the chain rule twice (to the sine and to the logarithm) in a single line, violating the rule that each step must change only one thing. The label "chain" is therefore incorrect for that step.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies the chain rule to both the sine term and the logarithm term in one line, violating the rule that each step must change only one thing. It should be split into two separate chain rule applications.
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.