Help with logarithmic differentiation

Possible derivation:
d/dx(x^(2 cos(x)))
Express x^(2 cos(x)) as a power of e: x^(2 cos(x)) = e^(log(x^(2 cos(x)))) = e^(2 log(x) cos(x)):
  =  d/dx(e^((2 cos(x)) log(x)))
Using the chain rule, d/dx(e^(2 log(x) cos(x))) = ( de^u)/( du) ( du)/( dx), where u = 2 log(x) cos(x) and ( d)/( du)(e^u) = e^u:
  =  (d/dx(2 cos(x) log(x))) e^(2 cos(x) log(x))
Express e^(2 log(x) cos(x)) as a power of x: e^(2 log(x) cos(x)) = e^(log(x^(2 cos(x)))) = x^(2 cos(x)):
  =  x^(2 cos(x)) d/dx(2 cos(x) log(x))
Factor out constants:
  =  2 d/dx(cos(x) log(x)) x^(2 cos(x))
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = cos(x) and v = log(x):
  =  log(x) d/dx(cos(x))+cos(x) d/dx(log(x)) 2 x^(2 cos(x))
The derivative of cos(x) is -sin(x):
  =  2 x^(2 cos(x)) (cos(x) (d/dx(log(x)))+-sin(x) log(x))
The derivative of log(x) is 1/x:
Answer: |  =  2 x^(2 cos(x)) (-(log(x) sin(x))+1/x cos(x))