Is Inverting a Derivative Always Possible?

[pellman]

Suppose that we have (on some domain) a 1 - 1 function y(x). So we can alternatively write x(y). Consider a point x_0 and let y_0 = y(x_0). Suppose

$$\frac{dy}{dx}(x_0) = f(x_0)$$

Is it always true that

$$\frac{dx}{dy}(y_0) = \frac{1}{f(x(y_0))}$$

? If not, under what conditions might it be false?

 

[HallsofIvy]

Suppose that we have (on some domain) a 1 - 1 function y(x). So we can alternatively write x(y). Consider a point x_0 and let y_0 = y(x_0). Suppose

$$\frac{dy}{dx}(x_0) = f(x_0)$$

Is it always true that

$$\frac{dx}{dy}(y_0) = \frac{1}{f(x(y_0))}$$

? If not, under what conditions might it be false?

As long as f is one-to-one and so has an inverse function, that is true. As usual, you can prove properties where you are treating the derivatve as if it were a fraction (here that dx/dy= 1/(dy/dx)) by going back before the limit of the difference quotient, using the fact that the difference quotient is a fraction and then taking the limit again.

 

[pellman]

Awesome. Thanks!