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(a) Suppose $ f $ is a one-to-one differentiable function and its inverse function $ f^{-1} $ is also differentiable. Use implicit differentiation to show that$ (f^{-1})'(x) = \frac {1}{f'(f^{-1}(x))} $provided that the denominator is not 0.(b) If $ f(4) = 5 $ and $ f'(4) = \frac {2}{3}, $ find $ (f^{-1})'(5). $
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01:33
Frank Lin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 5
Implicit Differentiation
Derivatives
Differentiation
Missouri State University
University of Michigan - Ann Arbor
University of Nottingham
Idaho State University
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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In this problem part, we first want to show that esprime of x is equal to 1 over on the failure of x. Now we know that f of inverse of x is a function, is 1 to 1 is equal to x. Now the keche derivative of both sides with respect to x, w c that an f prime of inverse of x times the rate of inner function times f inverse prime of x is equal to 1 point. So from the bee is elicited f. Inverse prime of x is equal to the 1 over f prime of inverse of x, and that is same as what is given in the problem statement. All right that was part a in part b. We are asked to find f inverse. Prime 5 point so we're going to use the equation that he just drive so f inverse prime of pi is equal to 1 over f prime f inverse of 5. What is that inverse of 5 point? Well, you know that f of 4 is equal to 4. If f of 4 is equal to 5, it means that inverse of 5 is equal to 4 point. So then we have 1 over f prime of 4 and prior is given. That is 2. Over 3 suit is 1 over 2 over 3. So we find the answer to be 3 over 2.
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