what is differentiation of |x|^|sinx| at x=-pi/4

Question

Basic differentiation 

 

Answer 1

I see no reason for the absolute value symbol around the base, because the base must not be negative.  Let f(x) = x^{sin x} = e^(lnx•sinx)

then differentiation will give f'(x) = x^{sin x}[(1/x)(sin x)+(ln x)(cos x)]

because (a) basically, derivative of exponential is exponential itself, except (b) you need the chain rule to be applied because the exponent (e-based) is not just x, but a product of two functions, so the chain-rule-factor is developed using the product rule to differentiate (lnx•sinx).

 

This can be evaluated at x = pi/4, but it seems to me that substituting negative pi/4 would not be kosher because the intrinsic use of ln x prohibits negative values.