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Chapter 5 Derivatives of Transcendental Functions 5-1 The Natural Logarithmic Function 5-4 Exponential Functions 5-5 Bases other than e and Applications 5-3 Inverse Functions and their Derivatives 5-8 Inverse Trig Functions 7-7 Indeterminate Forms and L’Hopital’s Rule 5.8 Derivatives of Inverse Trigonometric Functions We can use implicit differentiation to find: y sin 1 x sin y x dy cos y 1 dx dy 1 dx cos y d sin 1 x dx d d sin y x dx dx dy 1 dx 1 sin 2 y dy 1 dx 1 x2 sin 2 y cos2 y 1 cos2 y 1 sin 2 y cos y 1 sin 2 y But y 2 2 so cos y is positive. cos y 1 sin 2 y The derivatives of arcsecx and arctanx are found in a similar manner using the trig identity sec2 y 1 tan2 y . Derivatives of Inverse Trigonometric Functions d 1 du 1 sin u dx 1 u 2 dx d 1 du 1 tan u dx 1 u 2 dx d 1 du sec 1 u dx u u 2 1 dx Derivatives of the other Three There is a much easier way to find the other three inverse trigonometric functions using the following identities: It follows easily that the derivatives of the inverse cofunctions are the negatives of the derivatives of the corresponding inverse functions. Derivatives of Inverse Trigonometric Functions d 1 du 1 sin u dx 1 u 2 dx d 1 du 1 tan u dx 1 u 2 dx d 1 du sec 1 u dx u u 2 1 dx d 1 du 1 cos u dx 1 u 2 dx d 1 du cot 1 u dx 1 u 2 dx d 1 du csc 1 u dx u u 2 1 dx dy Examples: Find of the given functions. dx 1. y cos (3x ) dy 1 6x (6 x ) 2 2 dx (1 (3x ) 1 9 x4 1 2. y cot x dy 1 1 1 2 2 1 x x 1 dx 1 2 x dy 1 x (sec 1 x)(1) dx | x | x2 1 1 1 3. y x sec1 x 2 4. y arcsin x x 1 x 2 4. Solution y arcsin x x 1 x 2 1 2 1 2 2 y' x 2x 1 x 1 x 2 2 1 x 1 1 1 x 2 x 2 1 x 2 1 x 2 2 1 x 1 x 2 2 2 1 x 2 1 x 1 x 2 1 x 2 2 You Try… Find the derivative of the following functions. 1. f ( x) arctan x 1 f ( x) 2 x 1 x 2 1 2 x 1 x 2. g ( x) e arcsin( x) x 3. h( x) arc sec(e ) 2x 2e2 x 2x e e 2x 2 1 2 e4 x 1 You Try… 1 1 4. Given f ( x) cos ( x), find the equation 2 2 3 of the line tangent to this function at 2 , 8 . Closure Give the derivatives of all 6 inverse trig functions and a way for memorizing each.