Derivatives of trig functions

Having a ball teaching myself calculus for DSP...

I'm working with derivatives of trig functions. I'm trying to solve the following limit:

My first thought was to just set x to 0, since the equation would still remain valid. This is obviously wrong, when checking the answer in the back of the book, which gives this limit as 1/2. This question immediately follows a proof regarding the limit of sin(x)/x as x approaches zero always being 1, so I thought it might be helpful to look for ways to reduce this to something along those lines, but I'm just not seeing it. Can someone point me in the right direction?

Many thanks,

Brennon