Definition of the Derivative of a Function

Question

I'm supposed to show $$f'(x) = \lim_{h\to 0}\frac{f(x)-f(x-h)}{h} $$ I know I must use substitution at some point, but I don't think I can substitute $x - h$ for $x$. The definition I am using for $f^\prime(x)$ is $f^\prime(x) = \mbox{lim}_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} $,but I also could use $f^\prime(a) = \mbox{lim}_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}$.

Answer 1

Several things went wrong. First of all we have

$f'(a)=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$, if the limit exists.

I think that you have to show

$$f'(a)=\lim_{h \to 0}\frac{f(a)-f(a-h)}{h}.$$

Hint: $\frac{f(a)-f(a-h)}{h}=\frac{f(a-h)-f(a)}{-h}.$

Can you proceed ?

Answer 2

shouldn't the definition be $f^\prime(x) = \mbox{lim}_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$ ?