Derivative of \( \displaystyle \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \)
Problem 2.597 · hard Video
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 independently
Answer \( \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \left(4 \log{\left(4 x + 1 \right)} \cos{\left(4 x + 1 \right)} + \frac{4 \sin{\left(4 x + 1 \right)}}{4 x + 1}\right) \)
✓ Nihil obstat Lines: 6 proved, 1 checked by independent solution, 1 checked numerically. 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 | ≈ Checked numerically | claude-sonnet-5-5 script, run by sympy 1.14.0, 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 claude-sonnet-5-5 script, run by sympy 1.14.0: script ran past 60s ( checks/2.597/line2-5bf92ff1.py)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 independently | claude-sonnet-5-5 script, run by sympy 1.14.0, 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 claude-sonnet-5-5 script, run by sympy 1.14.0: replacing exp(sin(u)*log(u)) by u**sin(u) in line 6 gives line 7 exactly ( checks/2.597/line7-5d885709.py)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 two chain-rule derivatives (for sin and log) in a single step, violating the rule that each step must change only one thing. The label "chain" is also misleading because two separate chain applications occur simultaneously.qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, product rule, and standard derivatives in a logical sequence. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.
Senior review claude-sonnet-5-5, 2026-10-10: pass — The derivation is correct throughout. The rewrite a**b = exp(b*log a) and its reversal are definitional identities, and each chain, product and algebra step is correctly named.
gpt-oss:20b: dismiss — Step 5 applies a single rule (chain) to two terms, which the contract counts as one change, and the evaluated inner derivatives are part of the chain step; the label is correct.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — The derivation is correct throughout. The rewrite a**b = exp(b*log a) and its reversal are definitional identities, and each chain, product and algebra step is correctly named.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule, product rule, and standard derivatives in a logical sequence. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies two chain-rule derivatives (for sin and log) in a single step, violating the rule that each step must change only one thing. The label "chain" is also misleading because two separate chain applications occur simultaneously.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.