Derivative: Definition

How to find the derivative of a function at the point on the line by using the definition of the derivative: definition, 1 example, and its solution.

Definition

Definition

Think of two points on the graph y = f(x):
(a, f(a)) and (a + h, f(a + h)).

Draw a line that passes through these two points.

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

The change of x is h.

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

Then the line becomes the tangent of y = f(x) at x = a.

Limit of a Sequence

f'(a), f prime a, is the slope of y = f(x) at x = a.

So f'(a) = limh → 0 [f(a + h) - f(a)]/h.

This is the definition of the derivative f'(a).

Example

Example

Solution

f'(2) = limh → 0 [f(2 + h) - f(2)]/h.

f(x) = x2 - 3x + 1

Then f(2 + h) = (2 + h)2 - 3(2 + h) + 1.
And f(2) = 22 - 3⋅2 + 1.

(2 + h)2 = 22 + 2⋅2⋅h + h2 = 4 + 4h + h2
Square of a Sum

-3(2 + h) = -6 - 3h
Multiply a Monomial and a Polynomial

-[22 - 3⋅2 + 1] = -[4 - 6 + 1] = -4 + 6 - 1

Cancel 4 and -4.
Cancel -6 and +6.
And cancel +1 and -1.
+4h - 3h = +h

Then limh → 0 [h2 + h]/h.

Divide both of the numerator and the denominator by h.
Then limh → 0 [h + 1]/1.

Then limh → 0 [h + 1]/1 = [0 + 1]/1 = 1.

So f'(2) = 1.

Graph

This curve is the graph of y = f(x):
y = x2 - 3x + 1.

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