Difficult Derivative? I'm in a single-variable calculus course, in which we recently covered logarithmic differentiation.  The professor proved it that works when $f(x)>0$, and when $f(x)<0$.  I've been trying to find a way to derive that kind of function when $f(x)=0$, but I'm not sure if it's possible, or what.  I've thought of this example, that resists all my efforts to differentiate, but seems to be differentiable, (and even appears to have a value of zero).
Find 

$$f'\left(\frac{3\pi}{2}\right)\quad \rm where \quad f(x)=(\sin{x} + 1)^x .$$

Is there any way I can find this derivative (if it exists), beyond numerically computing the limit from the definition of the derivative? Or, vice versa, how can I prove that this derivative doesn't exist?
Thanks,
  Reggie

Graph of $f(x)$:

Graph of $f'(x)$:

See both the above graphs. 
$f(x)$ is actually not differentiable at $x= 1.5π$. The graph of $f'(x)$ at $x = 1.5π$ is a vertical asymptote. The function's second differential may say that it is increasing/decreasing at $x= 1.5π$ but the first derivative doesn't exist.
