Derivative of \( \displaystyle \left(2 x\right)^{\cos{\left(2 x \right)}} \)
Problem 2.1053 · hard
Differentiate \( \displaystyle f(x) = \left(2 x\right)^{\cos{\left(2 x \right)}} \).
- \[ \frac{d}{d x} \left(2 x\right)^{\cos{\left(2 x \right)}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\ln{\left(2 x \right)} \cos{\left(2 x \right)}} \]rewriteRewrite the base-exponent expression using the exponential and logarithm.≈ Checked numerically
- \[ = e^{\ln{\left(2 x \right)} \cos{\left(2 x \right)}} \frac{d}{d x} \ln{\left(2 x \right)} \cos{\left(2 x \right)} \]chainApply the chain rule.✓ Proved
- \[ = \left(\ln{\left(2 x \right)} \frac{d}{d x} \cos{\left(2 x \right)} + \cos{\left(2 x \right)} \frac{d}{d x} \ln{\left(2 x \right)}\right) e^{\ln{\left(2 x \right)} \cos{\left(2 x \right)}} \]productApply the product rule to the inner expression.✓ Proved
- \[ = \left(- 2 \ln{\left(2 x \right)} \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{x}\right) e^{\ln{\left(2 x \right)} \cos{\left(2 x \right)}} \]derivative algebraDifferentiate the individual terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(2 x\right)^{\cos{\left(2 x \right)}} \left(- 2 \ln{\left(2 x \right)} \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{x}\right) \]simplifyConvert back to the original base-exponent form.≈ Checked numerically
Answer \( \left(2 x\right)^{\cos{\left(2 x \right)}} \left(- 2 \ln{\left(2 x \right)} \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{x}\right) \)
✓ Nihil obstat Lines: 6 proved, 2 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left ((2*x)**cos(2*x)*(-2*x*log(2*x)*sin(2*x) + cos(2*x)) + (2*x*log(2*x)*sin(2*x) - cos(2*x))*exp(log(2*x)*cos(2*x)))/x; 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 x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left ((2*x)**cos(2*x)*(2*x*log(2*x)*sin(2*x) - cos(2*x)) + (-2*x*log(2*x)*sin(2*x) + cos(2*x))*exp(log(2*x)*cos(2*x)))/x; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x = 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 x = 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: pass — The solution correctly applies the exponential-logarithmic rewrite, chain rule, and product rule in distinct steps. All labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the exponential-logarithmic rewrite, chain rule, and product rule in distinct steps. All labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.