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lect116_11rev_f11

# lect116_11rev_f11 - 11.2 The derivative of the inverse...

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Friday, September 30 Lecture 11 : Derivatives of the inverse of functions . (Refers to Section 3.7 in your text) Students who have mastered the content of this lecture know about : The inverse of the trig functions, the derivative of arcsin x, arcos x, arctan x. Students who have practiced the techniques presented in this lecture will be able to : Find the inverse of the hyperbolic functions sinh x and cosh x, compute the derivative of the inverse of a function in a general way, the inverse of trig functions, and the inverse of hyperbolic functions. 11.1 The inverse trig functions. – We discussed the first three inverse trig functions earlier. We restate their definitions here along with the other three inverse trig functions as follows: Note that different authors may define the intervals for y differently. We will use the above definitions in this course. It is not essential for students to memorize these. It is useful to remember where to find them if you need them in the future.

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Unformatted text preview: 11.2 The derivative of the inverse functions – Consider the inverse function y = arcsin( x ). We can determine the derivative of this function by invoking the Chain rule. We proceed as follows: 11.2.1 All derivatives of the inverse trig functions are determined in this way : We prove that Proof : 11.3 Example – For the given function f ( x ) find f ′ ( x ). 11.4 Derivatives of inverse hyperbolic functions. Solution: 11.5 A general expression for the derivative of f − 1 ( x ). If y = f − 1 ( x ) then Proof: 11.5.1 Example – We find the derivative of tanh –1 ( x ). Recall from lecture 3 that tanh –1 ( x ) is only defined for | x | < 1. Also from lecture 3 recall the identity sech 2 x = 1 – tanh 2 x . Given: where | x | < 1. 11.5.2 Exercise – By mimicking the proof in example 11.4 show that the derivative of cosh –1 ( x ) is Derivatives of logarithmic functions log a x will be discussed in the next lecture....
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lect116_11rev_f11 - 11.2 The derivative of the inverse...

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