How can the chain rule prove the derivative of an inverse function?

jerometurner
Messages
6
Reaction score
0

Homework Statement



Assume f ^-1 (inverse) has a derivative. Use the chain rule to prove that
(f^-1)' (f(x)) = 1/f'(x)

Homework Equations



No real equations other than definition of chain rule.



The Attempt at a Solution



I'm not sure how to start other than with the definition of (f^-1) (f(x)) = x
 
Physics news on Phys.org
You have two ways of computing ((f^-1)' (f(x)))'. You could differentiate the composite function using the chain rule, or differentiate x = (f^-1)' (f(x)) directly. How do the answers compare?
 
Thanks for your help, I used the chainrule and figured it out.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

How Do You Prove the Multivariable Chain Rule?
Replies
4
Views
1K
Why Does the Multidimensional Chain Rule Lead to Euler's Theorem Proof?
Replies
2
Views
1K
Why Is the Chain Rule Not Used in Differentiating h(x) = 3f(x) + 8g(x)?
Replies
2
Views
2K
Derivatives and the chain rule
Replies
5
Views
2K
Derivative of the square root of the function f(x squared)
Replies
8
Views
1K
Inverse of a multivariate function
Replies
12
Views
3K
Proving piecewise function is k-differentiable
Replies
5
Views
2K
Non-Differentiable Function proof
Replies
8
Views
1K
Purpose of the derivative of the inverse function
Replies
7
Views
1K
Simplifying this derivative....
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top