5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS

By: Rafal, Paola , Mujahid

RECALL:

y=ex (exponential function)

LOGARITHM FUNCTION IS THE INVERSE

EX1: y=log4 x y=4x

Therefore y=ex y=log e x

IN THIS SECTION:

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

Key Points

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 ) use product rule

f’(x)= 2x e2x + 2 X2 e2x simplify terms

f’(x)=2x(1+x) e2x factor out 2x

b) f(x)= ex ÷ x f’(x)= (1) (ex )(x) – (ex )(1) ÷ X2 use quotient rule

f’(x)= x ex - ex ÷ X2 simplify terms

f’(x)= (x-1) (ex ) factor out ex

______

X2

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 Simplify

f’(x)= -2ex + xex÷ x3 Factor

f’(x)= (-2 + x) ex÷ x3 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

Example 4: The number, N, of bacteria in a culture at time t, in hours, is

N(t)=1000(30+e-).

a) What is the initial number of bacteria in the culture?

b) Determine the rate of change in the number of bacteria at time t.

c) What is happening to the number of bacteria in the culture as time passes?