CALCULUS · DERIVATIVES

The Derivative as Slope

Zoom in on a curve until it looks straight — that slope is the derivative.

15 min read 3 practice questions

The slope between two points on a curve is the average rate of change. Slide the second point closer and closer to the first, and the secant line becomes the tangent line. Its slope is the derivative.

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

The limit definition of the derivative

Secant → tangentInteractive

tangent secant

Shrink h to slide the second point toward the first. Watch the secant slope converge to the derivative.

Function

Point

x = a

1.0Step

h

1.50

f(x) = x^2

Secant slope3.500

$f'(a)$ 2.000

Worked example · From the definition

Use the definition to find

f'(x)

for

f(x) = x^2

1
$\frac{(x + h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h}$ .
2
Cancel $h$ : $2x + h$ .
3
Let $h \to 0$ : $f'(x) = 2x$ .

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

f(x) = x^2

, what is the slope of the tangent line at

x = 3

Find the average rate of change of

f(x) = x^2

from

x = 1

x = 4

f'(2) = 0

and

f'(x)

changes from negative to positive at

x = 2

, then at

x = 2

f

has a…

Numeric answers accept decimals or fractions like 3/4.