How to Find the Derivative of the Inverse Function f(x) for a Given Polynomial?

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In summary, the problem is asking for the derivative of the inverse function at x= -1, where the original function is f(x)= x^3- 3x^2- 1 for x\ge2. The inverse function changes the output of f to its input, and the derivative of the inverse function can be found using the theorem \frac{df^{-1}}{dx}(b)= \frac{1}{\frac{df}{dx}(a)} where f(a)= b.
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karush
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Let $f(x)={x}^{3}-3{x}^{2}-1, x\ge2$
$\text{find} \ {df}^{-1}/dx$
$ \text{at the point} \, \, x=-1=f(3)$
Not really sure what this is asking for
 
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[tex]f^{-1}[/tex] is the "inverse" of function f. If the function, f, changes "a" to "b" ([tex]f(a)= b[/tex]), then, if f has an inverse, [tex]f^{-1}[/tex] changes "b" to a- it "reverses" the function.
Was that what you didn't understand? It seems peculiar that you would be taking a Calculus course without having seen that before.

Here, [tex]f(x)= x^3- 3x^2- 1[/tex], for [tex]x\ge 2[/tex]. Notice that [tex]f(3)= 3^3- 3(3^2)- 1= 27- 27- 1= -1[/tex]. That is, f change 3 to -1 so the inverse function changes -1 to 3: [tex]f^{-1}(-1)= 3[/tex]. This problem asks you to find the derivative of [tex]f^{-1}(x)[/tex] at [tex]x= -1[/tex].

Your Calculus text should have, probably in the same section where you found this problem, a discussion of "inverse" functions as well as a statement (and probably a proof) of the theorem that [tex]\frac{df^{-1}}{dx}(b)= \frac{1}{\frac{df}{dx}(a)}[/tex] where f(a)= b.
 

FAQ: How to Find the Derivative of the Inverse Function f(x) for a Given Polynomial?

1. What does "242.q2.2 find df^{-1}/dx" mean?

This notation refers to finding the inverse of the derivative of a function, where f is the name of the function and x is the independent variable.

2. How do you find the inverse of the derivative of a function?

To find the inverse of the derivative, you will first need to take the derivative of the original function. Then, you can use the inverse function property to find the inverse of the derivative.

3. What is the purpose of finding the inverse of the derivative of a function?

Finding the inverse of the derivative can be useful in solving optimization problems and finding the rate of change of a function at a specific point.

4. Can the inverse of the derivative of a function be negative?

Yes, the inverse of the derivative can be negative. This indicates that the original function is decreasing at that specific point.

5. Are there any limitations to finding the inverse of the derivative of a function?

One limitation is that not all functions have an inverse. Additionally, the inverse of the derivative may not exist at certain points where the derivative is undefined, such as at sharp corners or discontinuities in the function.

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