By: Rafal, Paola , Mujahid
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y=ex (exponential function)

LOGARITHM FUNCTION IS THE INVERSE

EX1: y=log4 x
y=4x
 Therefore y=ex
y=log e x

The values of the derivative f’(x) are the same
as those of the original function y=ex

THE FUNCTION IS ITS OWN DERIVATIVE
f(x)=ex
f’(x)=ex
“e” is a constant called Euler’s number or the
natural number, where e= 2.718

The product, quotient, and chain rules can
apply to exponential functions when solving
for the derivative.
f(x)=e g(x)
Derivative of composite function:
f(x)=e g(x)
f’(x)= e g(x) g’(x)
by using the chain rule




f(x)= ex , f’(x)= ex
Therefore, y= ex has a derivative equal to
itself and is the only function that has this
property.
The inverse function of y=ln x is the
exponential function defined by y= ex.
Example 1:
Find the derivative of the following functions.
a) y= e 3x+2
b) y= ex2+4x-1
y’= g’(x) (f(x))
y’= g’(x) (f(x))
y’=(3)(e 3x+2 )
y’= (2x+4)(ex2+4x-1 )
y’= 3 e 3x+2
y’= 2(x+2)(ex2+4x-1 )

You can also use the product rule and
quotient rule when appropriate to solve.
 Recall:
Product rule:
f’(x)= p’(x)(q(x)) + p(x)(q’(x))

Quotient rule:
f’(x)=p’(x)(q(x)) – p(x)(q’(x))
___________________________
q(x)2
Example 2: Find the derivative and simplify
a)
f(x)= X2e2x
f’(x)= 2(x)(e2x ) + (X2 ) (2)(e2x )
f’(x)= 2x e2x + 2 X2 e2x
f’(x)=2x(1+x) e2x
use product rule
simplify terms
factor out 2x
b)
f(x)= ex ÷ x
f’(x)= (1) (ex )(x) – (ex )(1) ÷ X2
f’(x)= x ex - ex ÷ X2
f’(x)= (x-1) (ex )
______
X2
use quotient rule
simplify terms
factor out ex
Example 3: Determine the equation of the line
tangent to f(x) = ex÷x2 , where x= 2.
Solution:
Use the derivative to determine the slope of
the required tangent.
f(x) = ex÷x2
f(x)=x-2ex
Rewrite as a product
f’(x)= (-2x-3)ex + x-2 (1)ex
Product rule
f’(x)= -2ex ÷ x3 + ex÷ x2 Determine common denominator
f’(x)=-2ex ÷ x3 + xex÷ x3
f’(x)= -2ex + xex÷ x3
f’(x)= (-2 + x) ex÷ x3
Simplify
Factor
When x=2, f(2) = e2÷ 4. When x=2, f’(2)=0 . so
the tangent is horizontal because f’(2)=0.
Therefore, the equation of the tangent is f(2)
= e2÷ 4

a)
b)
c)
Example 4: The number, N, of bacteria in a
𝑡
culture at time t, in hours, is
N(t)=1000(30+e-30).
What is the initial number of bacteria in the
culture?
Determine the rate of change in the number of
bacteria at time t.
What is happening to the number of bacteria in
the culture as time passes?