# Calc03_9 - 29 ln u u d du a a a dx dx = → So far today we...

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3.9: Derivatives of Exponential and Logarithmic Functions Mt. Rushmore, South Dakota Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2007

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Look at the graph of x y e = The slope at x=0 appears to be 1. If we assume this to be true, then: 0 0 0 lim 1 h h e e h + - = definition of derivative
Now we attempt to find a general formula for the derivative of using the definition. x y e = ( 29 0 lim x h x x h d e e e dx h + - = 0 lim x h x h e e e h - = 0 1 lim h x h e e h - = 0 1 lim h x h e e h - = 1 x e = x e = This is the slope at x=0, which we have assumed to be 1.

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( 29 x x d e e dx =
x e is its own derivative! If we incorporate the chain rule: u u d du e e dx dx = We can now use this formula to find the derivative of x a

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( 29 x d a dx ( 29 ln x a d e dx ( and are inverse functions.) x e ln x ( 29 ln x a d e dx ( 29 ln ln x a d e x a dx (chain rule)
( is a constant.) ln a ( 29 x d a dx ( 29 ln x a d e dx ( 29 ln x a d e dx ( 29 ln ln x a d e x a dx ln ln x a e a ln x a a Incorporating the chain rule:

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Unformatted text preview: ( 29 ln u u d du a a a dx dx = → So far today we have: u u d du e e dx dx = ( 29 ln u u d du a a a dx dx = Now it is relatively easy to find the derivative of . ln x → ln y x = y e x = ( 29 ( 29 y d d e x dx dx = 1 y dy e dx = 1 y dy dx e = 1 ln d x dx x = 1 ln d du u dx u dx = → To find the derivative of a common log function, you could just use the change of base rule for logs: log d x dx ln ln10 d x dx = 1 ln ln10 d x dx = 1 1 ln10 x = ⋅ The formula for the derivative of a log of any base other than e is: 1 log ln a d du u dx u a dx = → u u d du e e dx dx = ( 29 ln u u d du a a a dx dx = 1 log ln a d du u dx u a dx = π 1 ln d du u dx u dx =...
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