# What is the limit definition of the derivative of the function y=f(x) ?

Mar 11, 2018

There are several ways of writing it. They all capture the same idea.

#### Explanation:

For $y = f \left(x\right)$, the derivative of $y$ (with respect to $x$) is

$y ' = \frac{\mathrm{dy}}{\mathrm{dx}} = {\lim}_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$

$f ' \left(x\right) = {\lim}_{\Delta x \rightarrow 0} \frac{f \left(x + \Delta x\right) - f \left(x\right)}{\Delta x}$

$f ' \left(x\right) = {\lim}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

$f ' \left(x\right) = {\lim}_{u \rightarrow x} \frac{f \left(u\right) - f \left(x\right)}{u - x}$