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AP Calculus BC 3.8 Derivatives of Inverse Trigonometric Functions Objective: able to calculate derivatives of the six inverse trigonometric functions. Recall from Section 1.6 Trigonometric Functions Inverse Trigonometric Functions Derivatives of Inverse Functions dy dx is differentiable If f is differentiable at every point of an interval I and is never zero on I , then f has an inverse and f −1 at every point on the interval f ( I ) . How are the derivatives of a function and its inverse related? 1. Find the derivative of arcsin. 2. Find if = √2 . 3. Find the derivative of arctangent. 4. A particle moves along the x-axis so that its position at any time tan . Find the velocity at t = 2. 5. Find the derivative of arcsecant. 6. Find if = . ≥ 0 is given by = Derivatives of the Other Three We could use the same techniques to find the derivatives of the other three inverse trigonometric functions: arccosine, arccotangent, and arccosecant, but it is much easier to think of the following identities. Inverse Function-Inverse Cofunction Identities cos −1 x = cot −1 x = csc −1 x = π 2 π 2 π 2 Calculator Conversion Identities − sin −1 x 1 sec −1 x = cos −1 x − tan −1 x cot −1 x = − sec −1 x 1 csc −1 x = sin −1 x π 2 − tan −1 x 7. Using the identities above, find the derivative of arccosine, arccotangent, and arccosecant. 8. Find the equation of the tangent line to the graph of = at x = 2. Rate yourself on how well you understood this lesson. I don’t get it at all I sort of get it I understand most of it but I need more practice I understand it pretty well I got it! 1 2 3 4 5 What do you still need to work on?