Derivative of \( \displaystyle x^{\cos{\left(x \right)}} \)

Problem 2.402 · hard

1. \[ \frac{d}{d x} x^{\cos{\left(x \right)}} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} e^{\ln{\left(x \right)} \cos{\left(x \right)}} \]
   rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
3. \[ = e^{\ln{\left(x \right)} \cos{\left(x \right)}} \frac{d}{d x} \ln{\left(x \right)} \cos{\left(x \right)} \]
   chainApply the chain rule for the exponential function.✓ Proved
4. \[ = \left(\ln{\left(x \right)} \frac{d}{d x} \cos{\left(x \right)} + \cos{\left(x \right)} \frac{d}{d x} \ln{\left(x \right)}\right) e^{\ln{\left(x \right)} \cos{\left(x \right)}} \]
   productApply the product rule to the exponent.✓ Proved
5. \[ = \left(- \ln{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) e^{\ln{\left(x \right)} \cos{\left(x \right)}} \]
   trigDifferentiate the trigonometric and logarithmic functions.✓ Proved
6. \[ = x^{\cos{\left(x \right)}} \left(- \ln{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) \]
   algebraSubstitute back the original expression for the exponential part.≈ Checked numerically
7. \[ = - x^{\cos{\left(x \right)}} \ln{\left(x \right)} \sin{\left(x \right)} + \frac{x^{\cos{\left(x \right)}} \cos{\left(x \right)}}{x} \]
   algebraDistribute the term x**cos(x).✓ Proved
8. \[ = x^{\cos{\left(x \right)} - 1} \cos{\left(x \right)} - x^{\cos{\left(x \right)}} \ln{\left(x \right)} \sin{\left(x \right)} \]
   simplifySimplify the first term by combining the powers of x.✓ Proved