Functions (mathematics)/Differentiation of a function

Definition

To differentiate a function (or to find a function's derivative), given a function $f(x)$, is to find the following limit. $f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

The derivative of a function, in Leibniz's notation, is denoted with an apostrophe in the function (read as f prime). For example, if there was a function $f(x)=x^2$, its derivative, or $f^{\prime }(x)$, can be found by plugging in $f(x)=x^2$ into the original limit.

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{(x+h{)}^2-x^2}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{(x^2+2xh+h^2)-x^2}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{2xh+h^2}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{h(2x+h)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }2x+h$

$f^{\prime }(x)=2x$

By finding the derivative of a function, the gradient of the tangent line of any point of a function can be found (unless the derivative does not exist).