I need help finding the derivative of an inverse function?

2013-03-01 4:30 pm

If S(x)= (inverse) f(x), then S'(x)=

That is an inverse, not an exponent of -1, the last person who tried to answer this didn't understand that.
How do I find the derivative of (inverse) f(x), when I don't know what f(x) is equal to? Are there any rules I can follow?

The back of the book explains,
f(S(x)) = x --> f ' (S(x)) x S'(x) = 1.

I don't recall learning any rule like this, will someone please explain this to me?

回答 (1)

An ESL Learner

2013-03-01 5:43 pm

✔ 最佳答案

If f and S are inverse functions, then
f(S(x)) = S(f(x)) = x.

Thus,
df(S(x))/dx = dx/dx = 1
= df(S(x))/dS(x) * dS(x)/dx (chain rule).

Therefore, dS(x)/dx = 1 / df(S(x))/dS(x).

參考: Definition of Inverse Function: http://www.regentsprep.org/Regents/math/algtrig/ATP8/inverselesson.htm Derivatives of Inverse Functions: http://oregonstate.edu/instruct/mth251/cq/Stage6/Lesson/inverseDeriv.html

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