Derivative of \( \displaystyle x^{\cos{\left(x \right)}} \)
Problem 2.402 · hard Video
Differentiate \( \displaystyle f(x) = x^{\cos{\left(x \right)}} \).
- \[ \frac{d}{d x} x^{\cos{\left(x \right)}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\ln{\left(x \right)} \cos{\left(x \right)}} \]rewriteRewrite the base and exponent using the exponential and logarithm.✓ Checked independently
- \[ = e^{\ln{\left(x \right)} \cos{\left(x \right)}} \frac{d}{d x} \ln{\left(x \right)} \cos{\left(x \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(\ln{\left(x \right)} \frac{d}{d x} \cos{\left(x \right)} + \cos{\left(x \right)} \frac{d}{d x} \ln{\left(x \right)}\right) e^{\ln{\left(x \right)} \cos{\left(x \right)}} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(- \ln{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) e^{\ln{\left(x \right)} \cos{\left(x \right)}} \]trigDifferentiate the trigonometric and logarithmic functions.✓ Proved
- \[ = x^{\cos{\left(x \right)}} \left(- \ln{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) \]algebraSubstitute back the original expression for the exponential part.✓ Checked independently
- \[ = - x^{\cos{\left(x \right)}} \ln{\left(x \right)} \sin{\left(x \right)} + \frac{x^{\cos{\left(x \right)}} \cos{\left(x \right)}}{x} \]algebraDistribute the term x**cos(x).✓ Proved
- \[ = x^{\cos{\left(x \right)} - 1} \cos{\left(x \right)} - x^{\cos{\left(x \right)}} \ln{\left(x \right)} \sin{\left(x \right)} \]simplifySimplify the first term by combining the powers of x.✓ Proved
Answer \( x^{\cos{\left(x \right)}} \left(- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) \)
✓ Nihil obstat Lines: 7 proved, 2 checked by independent solution. 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 independently | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left (x**cos(x)*(-x*log(x)*sin(x) + cos(x)) + (x*log(x)*sin(x) - cos(x))*exp(log(x)*cos(x)))/x; numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: derivatives agree after replacing exp(cos*log p) by p**cos (p>0) ( checks/2.402/line2-fcf519bd.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 x = 0 |
| 6 | ✓ Checked independently | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left (x**cos(x)*(x*log(x)*sin(x) - cos(x)) + (-x*log(x)*sin(x) + cos(x))*exp(log(x)*cos(x)))/x; numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: line 5 with exp(cos*log p) replaced by p**cos equals line 6, diff=0 ( checks/2.402/line6-a5d358e0.py)log is undefined for non-positive arguments undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 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: fail (error) — Step 5 applies two differentiation rules at once (trigonometric and logarithmic) but is labeled only as "trig". Each step must change only one thing and use a single rule name.qwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-08: pass — The derivation is correct: rewrite as exp, chain rule, product rule, evaluate derivatives, substitute back, distribute, combine powers. Line 2 and line 6 differ from their predecessors only by x**cos(x) = exp(cos(x)*log(x)), which holds for x>0, and domain restrictions are recorded separately.
gpt-oss:20b: dismiss — Step 5 just evaluates the basic derivatives left after the product rule (cos' and log'); 'trig' names a rule actually applied, so this is not a wrong label or a genuinely independent double change.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — The derivation is correct: rewrite as exp, chain rule, product rule, evaluate derivatives, substitute back, distribute, combine powers. Line 2 and line 6 differ from their predecessors only by x**cos(x) = exp(cos(x)*log(x)), which holds for x>0, and domain restrictions are recorded separately.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies two differentiation rules at once (trigonometric and logarithmic) but is labeled only as "trig". Each step must change only one thing and use a single rule name.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20
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.