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# Copy_of_3.8_notes - Page 46 Determine the derivative of the...

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Calculus Section 3.6 Page 44 3.8 Derivatives of Inverse Functions and Inverse Trigonometric Functions Example 1 Let f x ( ) = x 5 + 2 x - 1 . a) Since the point 1,2 ( ) is on the graph of f , why does it follow that the point 2,1 ( ) is on the graph of f - 1 ? b) Determine df - 1 dx 2 ( ) c) Develop a shortcut formula for f - 1 x ( ) ( ) f - 1 x ( ) ( ) = Example 2 Let f x ( ) = x 3 + x 2 + 1 a) Find f 1 ( ) and f 1 ( ) b) Find f - 1 3 ( ) and f - 1 ( ) 3 ( ) Derivative of the Arcsine Function Consider the function y = sin x What is the inverse of this function: Solve for y :

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Calculus Section 3.6 Page45 Use your graphing calculator to graph this inverse trigonometric function: What is the domain and range? Determine the derivative of the inverse sine function: Two new formulas: Arcsin x ( ) = sin - 1 x ( ) = Arcsin u ( ) = sin - 1 u ( ) = Example 3 Determine: d dx sin - 1 x 2 ( ) Derivative of the Arctangent Function Consider the function y = tan x What is the inverse of this function: Solve for y : Use your graphing calculator to graph this inverse trigonometric function: What is the domain and range?
Calculus Section 3.6

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Unformatted text preview: Page 46 Determine the derivative of the inverse tangent function: Two new formulas: Arctan x ( ) ′ = tan-1 x ( ) ′ = Arctan u ( ) ′ = tan-1 u ( ) ′ = Example 4 A particle moves along a line so that its position at any time t ≥ 0 is s t ( ) = tan-1 t . What is the velocity of the particle when t = 16? Example 5 a) Find an equation for the line tangent to the graph of y = tan x at the point -π 4 ,-1 . b) Find an equation for the line tangent to the graph of y = tan-1 x at the point -1,-π 4 . There are other inverse trigonometric functions . Here are their derivative formulas: Calculus Section 3.6 Page47 cos-1 u ( ) ′ = cot-1 u ( ) ′ = sec-1 u ( ) ′ = csc-1 u ( ) ′ = Example 5 Find dy dx for y = sin 1 x Assignment II–8 Pages 162 – 163 #1, 2, 3, 4, 6, 10, 11, 13, 16, 19 Additional Question: Given y = sin-1 x-1-x 2 , determine ′ y ....
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