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 Calculating the Inverse Rule (Posted on 2021-11-26)
Derivative David wants to find the rule for the derivative of f-1(x), the inverse of f(x). He knows the following derivative rules.

1. The Sum Rule: the derivative of (f+g)(x)=f(x)+g(x)
(f+g)'(x)=f'(x)+g'(x)

2. The Product Rule: the derivative of fg(x)=f(x)g(x)
(fg)'(x)=f'(x)g(x)+f(x)g'(x)

3. The Chain Rule: the derivative of (f*g)(x)=f(g(x))
(f*g)'(x)=f'(g(x))g'(x)

How can David find the Inverse Rule? What is the derivative of f-1(x)?

 No Solution Yet Submitted by Math Man Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 An Honest Attempt Comment 1 of 1
Let f^(-1)(x) = g(x)
Then, by definition of inverse functions, we must have:
f(g(x)) = x
Differentiating both sides w.r.t x, we have:
f'(g(x))*g'(x) = 1..... by Chain Rule
Or, g'(x) = 1/f'(g(x)
Or, d/dx (f^-1(x)) = 1/f'(f^-1(x)) ....(#)
The rhs of (#) therefore, does seem to be the sought for derivative. I have been unable to simplify this further.

Edited on November 27, 2021, 12:09 pm
 Posted by K Sengupta on 2021-11-26 21:27:46

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