The Manga Guide to Calculus

40 Chapter 1 Let’s Dierentiate a Function!

Calculating the Derivative of a Constant, Linear, or Quadratic

Function

1. Let’s find the derivative of constant function f(x) =

α

. The differential

coefficient of f(x) at x = a is

lim lim lim

εε

ε

ε

ε

αα

ε

→→

→

+

()

−

()

=

−

==

00

0

00

fa fa

Thus, the derivative of f(x) is f

′

(x) = 0. This makes sense, since our

function is constant—the rate of change is 0.

Note The differential coefficient of f(x) at x = a is often simply called the

derivative of f(x) at x = a, or just f

′

(a).

2. Let’s calculate the derivative of linear function f(x) =

α 

x +

β

. The deriva-

tive of f(x) at x = a is

lim lim lim

εε ε

ε

ε

αεβα β

ε

αα

→→ →

+

()

−

()

=