Suppose that $f:[a,b]\rightarrow \mathbb{R}$ is differentiable. Determine the value of the following limit. Prove that the limit is what you determined it to be: $$\lim\_{h\rightarrow 0}\frac{f(p+h)-f(p-h)}{h}.$$ We have that 

$$\begin{aligned}\lim\_{h\rightarrow 0}\frac{f(p+h)-f(p-h)}{h}& =\lim\_{h\rightarrow 0} \frac{f(p+h)-f(p)+f(p)-f(p-h)}{h} \\ & =\lim\_{h\rightarrow 0} \frac{f(p+h)-f(p)}{h}+\lim\_{h\rightarrow 0}\frac{f(p)-f(p-h)}{h} \\ & =f'(p)-\lim\_{h\rightarrow 0}\frac{f(p-h)-f(p)}{h}. \end{aligned}$$