Functions (mathematics)/Differentiation of a function

Definition

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

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 ${\displaystyle f(x)=x^{2}}$, its derivative, or ${\displaystyle f'(x)}$, can be found by plugging in ${\displaystyle f(x)=x^{2}}$ into the original limit.

${\displaystyle f'(x)=\lim _{h\to 0}{\frac {(x+h)^{2}-x^{2}}{h}}}$

${\displaystyle f'(x)=\lim _{h\to 0}{\frac {(x^{2}+2xh+h^{2})-x^{2}}{h}}}$

${\displaystyle f'(x)=\lim _{h\to 0}{\frac {2xh+h^{2}}{h}}}$

${\displaystyle f'(x)=\lim _{h\to 0}{\frac {h(2x+h)}{h}}}$

${\displaystyle f'(x)=\lim _{h\to 0}\ 2x+h}$

${\displaystyle f'(x)=2x}$

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).