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Derivative of a (exp) x.pdf

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3 <strong>Derivative</strong> <strong>of</strong> ax 2011.notebook<br />

April 07, 2011<br />

<strong>Derivative</strong>s <strong>of</strong> General Exponential Functions<br />

If y = a x prove that<br />

dy<br />

dx<br />

= a x ln a<br />

Pro<strong>of</strong>:<br />

if y = a x<br />

ln y = ln a x<br />

lny = x lna<br />

(differentiate both sides)<br />

but y = a x<br />

1


3 <strong>Derivative</strong> <strong>of</strong> ax 2011.notebook<br />

April 07, 2011<br />

Example 1: Differentiate each <strong>of</strong> the following.<br />

a)<br />

b)<br />

c)<br />

2


3 <strong>Derivative</strong> <strong>of</strong> ax 2011.notebook<br />

April 07, 2011<br />

Example 2: Determine the equation <strong>of</strong> the tangent line to<br />

f (x) = (2x 3 )(2 x 3<br />

) at x = 1.<br />

3


Pull<br />

3 <strong>Derivative</strong> <strong>of</strong> ax 2011.notebook<br />

April 07, 2011<br />

Application<br />

The size <strong>of</strong> a population <strong>of</strong> butterflies is given by the function<br />

where t is the time in days. Determine the rate <strong>of</strong> growth in the<br />

population after 5 days using the derivative.<br />

Population<br />

4


Pull<br />

3 <strong>Derivative</strong> <strong>of</strong> ax 2011.notebook<br />

April 07, 2011<br />

The number <strong>of</strong> bus riders from the suburbs to downtown per day is<br />

represented by N(t) = 1200(1.15) ­x , where x is the fare in dollars.<br />

What fare will maximize the total revenue.<br />

Revenue<br />

Fare<br />

5


3 <strong>Derivative</strong> <strong>of</strong> ax 2011.notebook<br />

April 07, 2011<br />

pg 240 #1­6, 8<br />

pg 245 #4 ­ 6, 9<br />

pg 248 #2, 6, 7, 9<br />

6

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