RedAssign2[1].4 - derivative of the inverse at Use the...

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Derivatives of Inverse Functions You can calculate the derivative of an inverse function at a point without determining the actual inverse function. The inverse of a function retains many of the properties of the original function. To derive the formula for the derivative of an inverse, start with a relationship you know. The composition of a function and its inverse is equal to x . You need to use implicit differentiation. Use the chain rule to differentiate both sides of that relationship. Isolate the derivative of the inverse by dividing. So if you know the value of the inverse at a point you can find the derivative of the inverse at that point. In this example, you know the function and the value of the inverse at . Your mission is to find the value of the
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Unformatted text preview: derivative of the inverse at . Use the formula that you learned above. The derivative of f ( x ) = 2 x + cos x is f ΄( x ) = 2 – sin x . Sine takes on values between –1 and 1, so the derivative lies between 1 and 3. It’s always positive, which means the function is increasing. Remember that increasing functions are invertible. Once you have found the derivative of the original function and verified that the function is invertible, all you have to do is plug into the formula. You have evaluated the derivative of the inverse of a function at a point, without determining the inverse itself! 1 © Thinkwell Corp. Elementary Functions and Their Inverses...
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