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# Derivatives of Exponential and Logarithmic Functions PowerPoint PPT Presentation

Derivatives of Exponential and Logarithmic Functions. Stewart Plus. Use the limit definition to find the derivative of e x. Find Find. Because. Use graphing calculator. The Derivative of e x. Therefore: The derivative of f ( x ) = e x is f ’( x ) = e x. Example 1.

Derivatives of Exponential and Logarithmic Functions

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## Derivatives of Exponential and Logarithmic Functions

Stewart Plus

### Use the limit definition to find the derivative of ex

Find

Find

Because

Use graphing calculator

### The Derivative of ex

Therefore: The derivative of f (x) = ex is f ’(x) = ex.

### Example 1

Find f’(x)

• f(x) = 4ex – 8x2 + 7x - 14

f’(x) = 4ex – 16x + 7

• f(x) = x7 – x5 + e3 – x + ex

f’(x) = 7x6 – 5x4 + 0 –1 + ex

= 7x6 – 5x4 –1 + ex

Remember that e is a real number, so the power rule is used to find the derivative of xe.

Also e2 7.389 is a constant, so its derivative is 0.

### Example 2

Find derivatives for

A) f (x) = ex / 2 f ’(x) = ex / 2

B) f (x) = 2ex +x2f ’(x) = 2ex + 2x

C) f (x) = -7xe– 2ex + e2f ’(x) = -7exe-1 – 2ex

### Review

is equivalent to

Domain: (0, ∞)

Range: (-∞, ∞)

Range: (0, ∞)

Domain: (-∞, ∞)

* These are inverse function. The graphs are symmetric with respect to the line y=x

* There are many different bases for a logarithmic functions. Two special logarithmic functions are common logarithm (log10x or log x) and natural logarithm (logex = ln x)

1)

2)

3)

4)

5)

Optional slide:

### Use the limit definition to find the derivative of ln x

Find

Property 2

Multiply by 1 which is x / x

Set s = h / x

So when h approaches 0, s also approaches o

Property 3

Definition of e

Property 4: ln(e)=1

### The Derivative of ln x

Therefore: The derivative of f (x) = ln x is f ’(x) =

Find y’ for

A)

B)

### More formulas

The derivative of f(x) = bx

is f’(x) = bx ln b

The derivative of f(x) = logb x

is f’(x) =

Proofs are on page 598

Find g’(x) for

A)

B)

### Example 5

An Internet store sells blankets. If the price-demand

equation is p = 200(0.998)x, find the rate of change of price

with respect to demand when the demand is 400 blankets

and explain the result.

p’ = 200 (.998)x ln(0.998)

p’(400) = 200 (.998)400 ln(0.998) = -0.18.

When the demand is 400 blankets, the price is decreasing about 18 cents per blanket

### Example 6

A model for newspaper circulation is C(t) = 83 – 9 ln t

where C is newspaper circulation (in millions) and t is the number of

years (t=0 corresponds to 1980). Estimate the circulation and find the

rate of change of circulation in 2010 and explain the result.

t = 30 corresponds to 2010

C(30) = 83 – 9 ln30 = 52.4

C(t)’ = C’(30) =

The circulation in 2010 is about 52.4 million and is decreasing at the rate of 0.3 million per year

### Example 7: Find the equation of the tangent line to the graph of f = 2ex + 6x at x = 0

Y = mx + b

f’(x) = 2ex + 6

m = f’(0) = 2(1) + 6 = 8

y=f (0) = 2(1) + 6(0) = 2

Y = mx + b

2 = 8(0) + b so b = 2

The equation is y = 8x + 2

### Example 8:Use graphing calculator to find the points of intersection

F(x) = (lnx)2 and g(x) = x