How Can I Prove the Reciprocal Derivative Identity?

[danago]

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

<br /> \frac{{dy}}{{dx}}\frac{{dx}}{{dy}} = 1<br />

Its nothing that I am 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 I've been able to do is show that it holds true for some examples that I've tried, but no solid proof.

Thanks in adavnce,
Dan.

 

[HallsofIvy]

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,
\frac{df^{-1}(y)}{dy}\frac{dy}{dx}= 1[/itex]<br /> Where I have &#039;let&#039; y= f(x). Since f^-1(y)= x, that is <br /> \frac{dx}{dy}\frac{dy}{dx}= 1[/itex]

 

[danago]

Ah, easy. Thanks very much for that