Lesson 10 derivative of exponential functions

DIFFERENTIATION OF EXPONENTIAL FUNCTIONS

OBJECTIVES:• apply the properties of exponential

functions to simplify differentiation;• differentiate functions involving exponential

functions; and• solve problems involving differentiation of

exponential functions.

The EXPONENTIAL FUNCTION

.ylogx as writtenbe also may ay function, c logarithmi

of inverse the is function l exponentia the Sincenumber. real a is x whereayby defined is 1,a

and 0a a, base withfunction l exponentia The

a

x

x

.

.

nmanama .1

n m if ,m-na1

n m if , 1n m if ,nma

nama .2

mnanma .3

nbnanab .4

nbnan

ba .5

0 a provided ,10a .6 n1ma

mn1anma .7

Laws of Exponents

xa .8 xloga

y x then aa if .9 yx

DIFFERENTIATION FORMULADerivative of Exponential Function The derivative of the exponential function for any given base and any differentiable function of u.

f(x)u where;dxdue )e(

dxd

f(x)u where;dxdualna )a(

dxd

uu

uu

:e base For

:abase given any For

A. Find the derivative of each of the following natural logarithm and simplify the result:

2x3exf .1

x21exg .2

x/12ex4xh .3 2

2

x3

x3

xe6x'f

x6ex'f

x212

2ex'g x21

x2ex

1ex4x'h x/12

x/12

EXAMPLE:

x21x21

x21ex'g

x21

x21e4x'h x/1

1x2e4x'h x/1

x21

x21ex'gx21

2yx2xxye .4

02y'yx1yx21y'xyxye

2y'xyyx2yxye'yxyxe

'xyy2xy2xye3y'yxye2xy

2xy2xye3yyxxye2xy'y

xye2y1x

xye2yxy21y'y

5x42x37y .5

5x42x3dx

d7ln5x42x37'y

4x67ln5x42x37'y

5x42x377ln2x32'y

2x34lnxh .6

2x34

2x34dxd

)x('h

2x34

2x3dxd4ln2x34

x'h

x64lnx'h 4lnx6x'h

2x34lnxh

4ln2x3xh

2xdx

d4ln3x'h

x24ln3x'h

4lnx6x'h

OR

3x2e1xelogxG .6

3elog1elogxG x2x

elog3e2eelog

1eex'G x2

x2

x

x

elog

3e1e1ee23eex'G x2x

xx2x2x

eloge3e1e

e2e23ex'G xx2x

xx2x2

eloge3e1e3e2e3x'G x

x2x

xx2

24 xx3 52xf .7

4224 x3xxx3 2dxd55

dxd2x'f

3x3xxx3 x122ln25x25ln52x'f4224

2lnx65ln52x2x'f 2xx3 24

x2lnx65ln52x'f 2x1x3 24

24 xx3 52xf

24 xx3 52lnxfln

24 xx3 5ln2lnxfln

5lnx2lnx3xfln 24

x25lnx42ln3xfx'f 3

5ln2lnx6x2xfx'f 2

5ln2lnx6x252)x('f 2xx3 24

x5ln2lnx652x'f 2x1x3 24

OR

yx53 .8 4yx

'yx4'y5ln53ln3 3yx

3ln3x415ln5'y x3y

15ln5

3ln3x4'y y

x3

A. Find the derivative and simplify the result.

1x3x2

3xg .1

22 xlnxexf .2

2eey .3 x3

x4

3x222 3xlogxh .4

2x5xG.1

2lnyxxeye.2 22yx

2X1xxH.3

1x2ey.4

1x2

x2x2 eelnxf.5

B. Apply the appropriate formulas to obtain the derivative of the given function and simplify.

EXERCISES:

Logarithmic DifferentiationOftentimes, the derivatives of algebraic functions which appear complicated in form (involving products, quotients and powers) can be found quickly by taking the natural logarithms of both sides and applying the properties of logarithms before differentiation. This method is called logarithmic differentiation.

1. Take the natural logarithm of both sides and apply the properties of logarithms.

2. Differentiate both sides and reduce the right side to a single fraction.

3. Solve for y’ by multiplying the right side by y.4. Substitute and simplify the result.

Steps in applying logarithmic differentiation.

Logarithmic differentiation is also applicable wheneverthe base and its power are both functions.

xxy if dx

dy Find .1

xlnxylnxlnyln x

Logarithmic differentiation is also applicable whenever the base and its power are both functions. (Variable to variable power.)

Example:

1xln1x1x'y

y1

xx y butyxln1'y

xxxln1'y

1x2ln1xyln

1x2lnyln 1x

1x1x2y if dx

dy Find .2

11x2ln21x2

11x'yy1

1x2ln1x21x2'y

y1

1-x12x y buty1x2ln1x21x2'y

1-x12x 1x2

1x2ln1x21x2'y

1-1-x12x 1x2ln1x21x2'y

2-x12x 1x2ln1x21x2'y

x5x6y .3

5x6lnxyln

5x6lny lnx

x2

15x6ln5x62

65x6

1xy'y1

12x

5x6x2

5x6ln5x6x6y'

x25x6ln

5x6x3y'

y1

5x6x2

5x6ln5x6x6y'y1

x5x6y buty

5x6x25x6ln5x6x6y'

x5x6

5x6x25x6ln5x6x6y'

1xx34y .4

x34ln1xyln

x34lny ln 1x

1x21x34ln3

x3411x'y

y1

1x2x34ln

x341x3'y

y1

1xx342

x34lnx341x6'yy1

1xx34y but y1xx342

x34lnx341x6y'

1xx341xx342

x34lnx341x6'y

11xx341x2

x34lnx341x6y'