# Derivative

How to find the derivative of a function at a given point (by using the definition of a derivative): definition, example, and its solution.

## Definition

The red line passes through two points
that are on f(x):
(a, f(a)) and (a + h, f(a + h)).

Then the slope of the red line is
[f(a + h) - f(a)] / [a + h - a]
= [f(a + h) - f(a)] / h.

Then, as h → 0,
(a + h, f(a + h)) gets close to (a, f(a)).

So the red line becomes the tangent line at x = a.

The derivative of f(x) at x = a, f'(a), means
the slope of the curve at x = a,
which is the slope of the tangent line at x = a.

So f'(a) is
the limit of [f(a + h) - f(a)] / h
as h → 0.

## Example

Use the definition of the derivative at x = 2.

(2 + h)2 = 22 + 2⋅2⋅h + h2

Square of a sum

Cancel the dark gray terms.

And add the h terms. (brown)

Divide the numerator and the denominator by h.

Then f'(2) = 1.

f'(2) = 1 means
the slope of the tangent line of f(x) at x = 2 is 1.

Or you can think as
the slope of the curve f(x) at x = 2 is 1.