How can the derivative of a differentiable function be expressed using limits?

[dgonnella89]

Homework Statement

Prove if f(x) is differentiable at x=a then $$f'(a)=lim_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{2h}$$

Homework Equations

I know that the derivative is defined as
$$f'(a)=lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$$

The Attempt at a Solution

Starting from the definition I used a known relation.
$$f'(a)=lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}=lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h} =lim_{h\rightarrow0}\left[\frac{f(a+h)-f(a-h)}{h}+\frac{f(a-h)-f(a)}{h}\right]$$

I'm not sure how to decompose anymore from here. Any help you could give would be greatly appreciated.

 

[tiny-tim]

Hi dgonnella89!

Hint: put k = -h in the definition of f'(a).