Derivative

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 derivative of a function at a point is the slope of the curve at the point, which is the slope of the tangent line at the point.

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

Find f'(2) by using the definition of the derivative. f(x) = x^2 - 3x + 1

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.