# 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} = 2^{2} + 2⋅2⋅*h* + *h*^{2}

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.