I know the inverse of a function is found in two steps.(adsbygoogle = window.adsbygoogle || []).push({});

Isolate the independent variable and then switch the variables like this:

[tex][y = x^{3} +1] = [x = \sqrt[3]{y - 1}][/tex]

Then switch the variables to get: [tex] y = \sqrt[3]{x-1}[/tex]

However, when it comes to finding the derivative of the inverse of a function, is it true that the inverse of the function does not actually have to be found prior to differentiating?

For instance, I have this function: [tex]y=x^{5} + x + 1[/tex]

If all I do is switch the variables like so: [tex] x=y^{5}+y+1[/tex]

and then differentiate implicitly like so: [tex]\frac{d}{dx}[x] = \frac{d}{dx}[y^{5}+y+1] [/tex]

[tex]= [1= (5y^4 +1)\frac{dy}{dx}] = [\frac{dy}{dx} = \frac{1}{5y^{4}+1}] [/tex]

Two questions:

1) Is this the/a correct way to differentiate the inverse of a function?

2) If not correct, do I have to first find the inverse and then differentiate?

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# Finding the derivative on the inverse of a function

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