Calc03_8 - 3.8 Derivatives of Inverse Trig Functions Lewis...

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3.8 Derivatives of Inverse Trig Functions Lewis and Clark Caverns, Montana Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1993
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( 29 2 0 f x x x = We can find the inverse function as follows: 2 y x = Switch x and y . 2 x y = x y = y x = 2 y x = y x = 2 df x dx = At x = 2 : ( 29 2 2 2 4 f = = ( 29 2 2 2 4 df dx = = 4 m = ( 29 2,4 ( 29 1 f x x - = ( 29 1 1 2 f x x - = 1 1 2 1 2 df x dx - - = 1 1 2 df dx x - = To find the derivative of the inverse function:
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( 29 2 0 f x x x = 2 y x = y x = 2 df x dx = At x = 2 : ( 29 2 2 2 4 f = = ( 29 2 2 2 4 df dx = = 4 m = ( 29 2,4 ( 29 1 f x x - = 1 1 2 df dx x - = ( 29 1 1 1 1 4 2 2 4 2 4 df dx - = = = At x = 4 : ( 29 1 4 4 2 f - = = ( 29 4,2 1 4 m = Slopes are reciprocals.
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2 y x = y x = 4 m = ( 29 2,4 ( 29 4,2 1 4 m = Slopes are reciprocals. Because x and y are reversed to find the reciprocal function, the following pattern always holds: Derivative Formula for Inverses: df dx df dx x f a x a - = = = 1 1 ( ) evaluated at ( ) f a is equal to the reciprocal of the derivative of ( ) f x evaluated at . a The derivative of 1 ( ) f x -
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A typical problem using this formula might look like this: Given: ( 29 3 5 f = ( 29 3
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