Derivative of a (exp) x.pdf
Derivative of a (exp) x.pdf
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 />
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