A problem with logarithmic differentiation

Question

I'm solving a problem to find if h(x) = (a^|x|)sgn(x) is increasing or decreasing (taking a>1) for all real values of x. For x>0 and for x=0, I have found that f'(x) >= 0.. But for x<0, h(x)=-(a^{-x}) and I can't figure out how to differentiate this.. If I apply logarithmic differentiation I get log(h(x))= (-x)(log(-a)) and already I have log of a negative number.. How do I differentiate this now? When I graph h(x) for x<0, I can see that it is differentiable, but I can't figure out its derivative..

Answer 1

Hint. You have $$ h(x)=-a^{-x}, \quad x<0, $$ if you write it as $$ h(x)=-e^{-x \ln a}, \quad x<0, $$ maybe it becomes easier to apply the chain rule: $$ \left(e^{u(x)} \right)'=u'(x)\cdot e^{u(x)}. $$