3.9: Derivatives of Exponential and Log FunctionsObjective:

To find and apply the derivatives of exponential and logarithmic functions

QUICK REVIEW…..Properties of logs and exponential functions

xa

axa

yx

x

x

a

a

log.3

loglog.2

;log.1 xa y

xe

axa

yx

x

x

ln.3

lnln.2

;ln.1 xe y

Derivative of e x

Important limit:

Proof:

11

lim0

h

eh

h

xedx

d

If u is a differentiable function of x, then

Examples: Find y’.

'ueedx

d uu

x

xx

x

ey

ey

ey

1

2

sin

5

3

.3

.2

.1

Derivative of a x

Assume a is positive, and different from 1

Use properties of logs to write ax in terms of ex:

)( ln

ln

ln

ax

axx

ax

edx

d

ea

eax

If u is a differentiable function of x, then:

Examples: Find the derivative.

'ln uaaadx

d uu

2

7.3

4.2

2.1

sin

3

x

x

x

y

y

y

At what point on the function y=4t-5 does the tangent line have a slope of 15?

Derivative of ln x :y=ln x

ey=x

Use implicit to differentiate:

If u is a differentiable function of x and u>0, then:

Examples: Find dy/dx.

'1

ln uu

udx

d

)ln(tan.3

)24ln(.2

)3ln(.1

23

xy

xxxy

xy

Derivative of loga x:

a

xxa ln

lnlog Change of Base formula

a

x

dx

d

ln

ln

If u is a differentiable function of x and u>0, then:

Examples: Find y’:

'ln

1log u

auu

dx

da

, for a >0, a ≠ 1

xy

xy

xy

cos2

3

3

2log.3

log.2

log.1

Find the derivative of the following functions.

ty

xy

xxey

t

x

xx

2

3

1

)tan3(

3.3

1sin2.2

)cos(sin.1

2

Find an equation of the tangent line to the graph of the function at the given point.

)0,1(,ln.2

)1,5(),2log(.1

xey

xy

x

Power Rule for Arbitrary Real Powers

If u is a positive differentiable function of x and n is any real number, then un is a differentiable function of x and

3

1

:

'

xyExample

unuudx

d nn

Logarithmic DifferentiationFind the derivative:

y=(x-2)x+1 ← Notice x in base and exponent!! No rule for this!

FIND DY/DXxxy 2