Difference in definition of differentiation Okay so quite often I see two different definitions of differentiation and I want to know when it is appropriate to use each one.

$$\lim_{h \to 0} \frac{f(x_{0}+h)-f(x_{0})}{h}$$
and
$$\lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0}$$

Why are their to definitions, how are they different and when do you use each one ?
It is almost routine in mathematics than one thing can be expressed into two different visible forms without any problem and in most cases it is a matter of convenience to prefer one form over another. For example we can write $1/2$ as $0.5$ also and it does not matter which form we choose.

In a similar manner both the definitions of derivative are equivalent. It should be easy to prove that $\lim_{x \to a}g(x) = \lim_{h \to 0}g(a + h)$ and choosing $a = x_{0}$ and $g(x) = \dfrac{f(x) - f(x_{0})}{x - x_{0}}$ we see that $$\lim_{x \to x_{0}}g(x) = \lim_{h \to 0}g(x_{0} + h)$$ or $$\lim_{x \to x_{0}}\frac{f(x) - f(x_{0})}{x - x_{0}} = \lim_{h \to 0}\frac{f(x_{0} + h) - f(x_{0})}{h}$$
This is essentially the mean value theorem.
$$\lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0}$$

Note how similar this formula is to calculating the slope of a line.

$$m = \frac{y_2-y_1}{x_2-x_1}$$

Also note that $x-x_0$ is essentially the h in the definition of the derivative, as $\lim_{x\to x_0} x-x_0$ goes to zero, just as h does. The mean value theorem can be derived by approximating f'(x) with secant lines. Let $x$ go to $x_0$ and it will yield tangent lines. However, both should yield the same answer. They are two ways of expressing the same idea. I would recognize the first one as the true definition of a derivative, but the mean value theorem is also useful for differentiation.