# Reciprocal Derivative Identity

1. Jul 15, 2007

### danago

Hi. I was just wondering, how can i prove the following identity:

$$\frac{{dy}}{{dx}}\frac{{dx}}{{dy}} = 1$$

Its nothing that im required to know, but i was just curious, so for all i know, it may be way out of anything that i can mathematically comprehend.

The best ive been able to do is show that it holds true for some examples that ive tried, but no solid proof.

Dan.

Last edited by a moderator: Jul 15, 2007
2. Jul 15, 2007

### HallsofIvy

Staff Emeritus
It should be easy using the chain rule. If y= f(x) and f is invertible, then
x= f-1(y), so that f-1(f(x))= x. Differentiating both sides of that with respect to x,
[tex]\frac{df^{-1}(y)}{dy}\frac{dy}{dx}= 1[/itex]
Where I have 'let' y= f(x). Since f-1(y)= x, that is
[tex]\frac{dx}{dy}\frac{dy}{dx}= 1[/itex]

3. Jul 15, 2007

### danago

Ah, easy. Thanks very much for that