The Manga Guide to Calculus

140 Chapter 4 Let’s Learn Integration Techniques!

Summary of Exponential and Logarithmic Functions

u

′

( )

( )

f x

f x

is thought to be the growth rate.

v y = f(x) which satisfies

′

( )

( )

f x

f x

= 1 is the function that has a constant

growth rate of 1.

This is an exponential function and satisfies

′

( )

=

( )

f x f x

w If the inverse function of y = f(x) is given by x= g(y), we have

′

( )

=g y

y

1



 If we define g(

α

), we can find the area of h(y) =

1

y

,

g

y

dy

α

α

( )

=

∫

1

1

The inverse function of f(x) is the function that satisfies  and g(1) = 0.

y

z

e

z

y

=

1

Area = 1

1



e is an irrational number

that is about 2.7178.

We define e (the base of the

natural logarithm) as