Math 1140, WS #15, Derivatives of Logarithmic and Exponential Functions PDF

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Worksheet 15...

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MATH 1140 – WS #15 3.9 DERIVATIVES of LOGARITHMIC and EXPONENTIAL FUNCTIONS. LOGARITHMIC DIFFERENTIATION 1.

Differentiate the following functions: a)

y  5 ln x

b)

y  ln (5 x)

c)

y  ln (x 5 )

d)

y  ( ln x) 5  ln 5 x

e)

y  ln (2 x  3)

f)

y

g)

 1  y  ln    2x  3 

1 ln (2x  3)

2

2.

Math 1140 – WS #15 - Derivatives of Logarithmic and Exponential Functions. Logarithmic Differentiation h)

y  ln 3 ( x 2 )

i)

y

j)

y  ln

k)

1 y  ln ( x 1  2 x ) , x  . 2

x  ln x 1 x

x2 1 x2  1

Find x such that h' ( x)  0 if: a)

h (x ) 

ln x x

b)

h(x) 

1 ln ( x 2  5)

3

3.

4.

Math 1140 – WS #15 - Derivatives of Logarithmic and Exponential Functions. Logarithmic Differentiation Differentiate: a)

f (x )  x 2 ln (x  2)

b)

g ( x)  x ln (x 2  2)

c)

z

d)

h( x)  4 tan (ln x) .

x  ln x x

Find an equation of the line tangent to y  ln (cos x) at x 

 4

.

4

5.

6.

Math 1140 – WS #15 - Derivatives of Logarithmic and Exponential Functions. Logarithmic Differentiation Differentiate the following functions with respect to x: a)

y  e3 x

c)

y

1 (0.5) 2x

b)

y  e

x

d)

y  2x

2

Find the derivative of the following functions: a)

f (x )  e2 x x

c)

F ( x )  x( 7 x )

d)

h( x)  ( x 2  7 x  4) e7 x

2

b)

g ( x )  5x  x  e 5

3

5

7.

Math 1140 – WS #15 - Derivatives of Logarithmic and Exponential Functions. Logarithmic Differentiation 1 e  e t

e)

y

f)

y  (3 x  2) 9 ( e2x  3) 7

t

Use logarithmic differentiation to find the derivative of the given function: a)

y  x cos x

b)

y xx

c)

y  x ln x

d)

y  (tan x) x

1

6

Math 1140 – WS #15 - Derivatives of Logarithmic and Exponential Functions. Logarithmic Differentiation ANSWERS:

1.

5 , x 5 c) y '  , x 2 e) y '  , 2x  3

1 , x 5(ln x )4 d) y'  , x

a) y ' 

g) y'  

b) y' 

f) y '  

2 , 2x  3

h) y' 

2 , (2 x  3) ln 2 (2 x  3)

6 ln 2 x 2 , x

2x 2x  1  x ln x , j) y '  4 . 2 x (1  x) x 1 1 3x  1 since y  ln ( x 1  2 x )  ln x  ln (1  2 x) . y'  2 x(2 x  1)

i) y '  k)

2.

a) x  e ,

3.

a) f ' ( x)  2 x ln ( x  2)  c) z' 

4.

y

b) when x  0 .

2x 2 , x2  2 4 sec 2 (ln x ) d) h' ( x)  . x

x2 , x2

b) g' ( x)  ln ( x2  2) 

2x  2  x  ln x , 2x x

 ln 2  (x  ) 2 4

5. a) y'  3 e3 x ;

b) y'  e  x ;

6. a) f ' (x )  2(1 x )e2 x x ; 2

y  0.35  ( x  0.79) .

or

2

c) y'  2(0.5) 2x ln( 0.5) ; d) y '  2x 2 x ln 2 .

b) g ' (x )  5x ln 5  5x 4  0 ;

d) h' ( x)  (7 x 2  47 x  21) e7 x ;

e) y '  

c) F ' (x )  (1  x ) 7 x ln 7;

et  e t e t  e t  ; (e t  e  t ) 2 e 2 t  e 2 t  2

f) y ' (3x  2)8 (e 2x  3) 6 [27(e 2x  3)  14e 2x (3x  2)]

7.

y ' x

c) y ' 

2 ln x ln x x , x

1 x

1

2 1  ln x  x ( 1 ln )   x x b) y'  x  ,  2  x   x sec 2 x  d) y '  ( tan x )x  ln(tan x )  . tan x  

cos x  (sin x) ln x) , ( x

a) .

cosx...