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 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.

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Derivatives of Exponential and Logarithmic Functions

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Derivatives of exponential and logarithmic functions

Derivatives of Exponential and Logarithmic Functions

Stewart Plus


Use the limit definition to find the derivative of e x

Use the limit definition to find the derivative of ex

Find

Find

Because

Use graphing calculator


The derivative of e x

The Derivative of ex

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


Example 1

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


Example 2

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

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)


Review properties of ln

Review: properties of ln

1)

2)

3)

4)

5)


Use the limit definition to find the derivative of ln x

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

The Derivative of ln x

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


Example 3

Example 3

Find y’ for

A)

B)


More formulas

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


Example 4

Example 4

Find g’(x) for

A)

B)


Example 5

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

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 2e x 6x at x 0

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

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

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

On your calculator, press Y=

Type in the 2 functions above for Y1 and Y2

Press ZOOM, 6:ZStandard

To have a better picture, go back to ZOOM, 2: Zoom In

*Now, to find the point of intersection (there is only 1 in this problem), press 2ND, TRACE then 5: intersect

Play with the left and right arrow to find the linking dot, when you see it, press ENTER, ENTER again, then move it to the intersection, press ENTER. From there, you should see the point of intersection

(.49486641, .49486641)


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