# Is the derivative equal to one over the derivative of the inverse?

1. Feb 15, 2005

### Crosson

Is it true that (dy/dx) = 1/(dx/dy) ? Is it still true if these are partial derivatives of a function of multiple variables? (All this, assuming that the inverse function exists)

Can anybody prove it in terms of the definition of a derivative? or any sort elementary proof at all? (I can almost convince my self it is true with Linear Algebra).

I haven't seen it proven in any books, but it is true for all of the functions I know of. Can I get a definitive answer?

2. Feb 15, 2005

### matt grime

yes, dy/dx= 1/(dx/dy), when both are defined.

It doesn't make sense to ask about it for partial derivatives in the manner you do.

Suppose that y=f(x) implicitly defines y as a function of x. differentiate both sides wrt y

1 = df(x)/dy

apply the chain rule

1= (dx/dy) (df/dx)

or

1= (dx/dy)(dy/dx)

For partials you'd need to be more specific, are x and y both variables of this functions, or is, y=f(x,z,w...) the thing you're after? You can apply what I did above in the partial case for the second type.