Definition of the Derivative
Table of Contents
$$ \frac{d}{dx}f(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} $$
Where
- \(f(x)\) is a function,
- \(x\) is the independent variable, and
- \(h\) is an increment that we take to 0 in the limit.
Sources #
Example #
Find the derivative of \(f(x) = x^2\) at \(x=2\) using the definition of the derivative.
We can plug in everything into the definition to get the answer:
$$ \scriptsize \frac{d}{dx}f(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\to 0}\frac{(x+h)^2-x^2}{h} = \lim_{h\to 0}\frac{x^2+2xh+h^2-x^2}{h} $$
$$ \scriptsize = \lim_{h\to 0}\frac{2xh+h^2}{h} = \lim_{h\to 0}2x+2h = 2x $$
Evaluated at \(x=2\), we get \(4\).