Derivative of \( \displaystyle \left(4 x + 2\right)^{\sin{\left(4 x + 2 \right)}} \)
Problem 2.937 · hard
Differentiate \( \displaystyle f(x) = \left(4 x + 2\right)^{\sin{\left(4 x + 2 \right)}} \).
- \[ \frac{d}{d x} \left(4 x + 2\right)^{\sin{\left(4 x + 2 \right)}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\ln{\left(4 x + 2 \right)} \sin{\left(4 x + 2 \right)}} \]rewriteRewrite the power function using the exponential identity.≈ Checked numerically
- \[ = e^{\ln{\left(4 x + 2 \right)} \sin{\left(4 x + 2 \right)}} \frac{d}{d x} \ln{\left(4 x + 2 \right)} \sin{\left(4 x + 2 \right)} \]chainApply the chain rule to the exponential function.✓ Proved
- \[ = \left(\ln{\left(4 x + 2 \right)} \frac{d}{d x} \sin{\left(4 x + 2 \right)} + \sin{\left(4 x + 2 \right)} \frac{d}{d x} \ln{\left(4 x + 2 \right)}\right) e^{\ln{\left(4 x + 2 \right)} \sin{\left(4 x + 2 \right)}} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(4 \ln{\left(4 x + 2 \right)} \cos{\left(4 x + 2 \right)} + \frac{4 \sin{\left(4 x + 2 \right)}}{4 x + 2}\right) e^{\ln{\left(4 x + 2 \right)} \sin{\left(4 x + 2 \right)}} \]chain algebraDifferentiate the trigonometric and logarithmic terms using the chain rule. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(4 x + 2\right)^{\sin{\left(4 x + 2 \right)}} \left(4 \ln{\left(4 x + 2 \right)} \cos{\left(4 x + 2 \right)} + \frac{4 \sin{\left(4 x + 2 \right)}}{4 x + 2}\right) \]simplifySubstitute the original function back into the expression.≈ Checked numerically
Answer \( \left(4 x + 2\right)^{\sin{\left(4 x + 2 \right)}} \left(4 \ln{\left(4 x + 2 \right)} \cos{\left(4 x + 2 \right)} + \frac{4 \sin{\left(4 x + 2 \right)}}{4 x + 2}\right) \)
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 2*(2*(2*x + 1)*log(4*x + 2)*cos(4*x + 2) + sin(4*x + 2))*((4*x + 2)**sin(4*x + 2) - exp(log(4*x + 2)*sin(4*x + 2)))/(2*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 + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 2 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*(2*(2*x + 1)*log(4*x + 2)*cos(4*x + 2) + sin(4*x + 2))*(-(4*x + 2)**sin(4*x + 2) + exp(log(4*x + 2)*sin(4*x + 2)))/(2*x + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 4*x + 2 = 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 + 2 = 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 differentiations (for sin and log) in a single line, violating the rule that each step must change only one thing. The label "chain" is also misleading because the product rule was already applied in step 4.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — Step 5 applies two chain-rule differentiations (for sin and log) in a single line, violating the rule that each step must change only one thing. The label "chain" is also misleading because the product rule was already applied in step 4.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — Step 5 applies two chain rule applications in a single step (d/dx sin and d/dx log). Each rule application must be a separate step; the step should be split into two distinct 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 2026-09-27 with SymPy 1.14.0.