Derivatives of Exponential and

Logarithmic Functions

Michael Freeze

MAT 151UNC Wilmington

Summer 2013

1 / 23

Section 4.4 :: Derivatives of Exponential Functions

2 / 23

Derivative of ex

d

dx(ex) = ex

Example

Find the derivative of y = 5ex .

3 / 23

Derivative of ex

d

dx(ex) = ex

Example

Find the derivative of y = ex+7.

4 / 23

Derivative of ex

d

dx(ex) = ex

Example

Find the derivative of y = x ex .

5 / 23

Derivative of ex

d

dx(ex) = ex

Example

Find the derivative of y = ex

x+1.

6 / 23

Derivative of ax

For any positive constant a 6= 1,

d

dx(ax) = (ln a) ax

Example

Find the derivative of f (x) = 2x .

7 / 23

Chain rule for exponential functions

d

dx

(eg(x)

)= eg(x) · g ′(x)

Example

Find the derivative of y = e3x .

8 / 23

Chain rule for exponential functions

d

dx

(eg(x)

)= eg(x) · g ′(x)

Example

Find the derivative of y = ex2−x .

9 / 23

Chain rule for exponential functions

d

dx

(eg(x)

)= eg(x) · g ′(x)

Example

Find the derivative of y = −3e−x2+3.

10 / 23

Chain rule for exponential functions

d

dx

(eg(x)

)= eg(x) · g ′(x)

Example

Find the derivative of y = (x2 + x)e2x−1.

11 / 23

Pollution Concentration

The concentration of pollutants (in grams per liter)in the east fork of the Big Weasel River isapproximated by

P(x) = 0.04e−4x ,

where x is the number of miles downstream from apaper mill that the measurement is taken.

Find the following values.

(a) The concentration of pollutants 0.5 mile downstream

(b) The concentration of pollutants 1 mile downstream

(c) The concentration of pollutants 2 miles downstream

12 / 23

Pollution Concentration

The concentration of pollutants (in grams per liter)in the east fork of the Big Weasel River isapproximated by

P(x) = 0.04e−4x ,

where x is the number of miles downstream from apaper mill that the measurement is taken.

Find the rate of change of concentration with respect to dis-tance for the following distances.

(d) 0.5 mile

(e) 1 mile

(f) 2 miles

13 / 23

Section 4.5 :: Derivatives of Logarithmic Functions

14 / 23

Derivative of ln x

d

dx[ln x ] =

1

x

15 / 23

Chain Rule for Logarithmic Functions

d

dx[ln g(x)] =

g ′(x)

g(x)

Example

Find the derivative of y = ln(8x).

16 / 23

Chain Rule for Logarithmic Functions

d

dx[ln g(x)] =

g ′(x)

g(x)

Example

Find the derivative of y = ln(x2 − 3x + 1).

17 / 23

Chain Rule for Logarithmic Functions

d

dx[ln g(x)] =

g ′(x)

g(x)

Example

Find the derivative of y = x ln x .

18 / 23

Chain Rule for Logarithmic Functions

d

dx[ln g(x)] =

g ′(x)

g(x)

Example

Find the derivative of y = ln(√

4x − 3).

19 / 23

Chain Rule for Logarithmic Functions

d

dx[ln g(x)] =

g ′(x)

g(x)

Example

Find the derivative of y = ln(

13x−1

).

20 / 23

Logarithmic Differentiation

Find the derivative of

y = (x + 4)4 (3x − 1)2.

21 / 23

Logarithmic Differentiation

Find the derivative of

y =

√3x − 1

(x2 + 2x)3.

22 / 23

Field Metabolic Rate

The field metabolic rate (FMR), or the total energyexpenditure per day in excess of growth, can becalculated for pronghorn fawns using Nagy’sformula,

F (x) = 0.774 + 0.727 log x ,

where x is the mass (in grams) of the fawn andF (x) is the energy expenditure (in kJ/day).Source: Animal Behavior.

(a) Determine the total energy expenditure per dayin excess of growth for a pronghorn fawn thatweighs 25,000 g.

(b) Find F ′(125, 000) and interpret the result.23 / 23