# derivatives of inverse functions

• Nov 21st 2009, 07:33 PM
oblixps
derivatives of inverse functions
this topic probably isn't supposed to be hard but i find it very confusing. my teacher went over it very briefly if not at all and my book doesn't help explain either. so it's my understanding that if you let y = f^-1 (x), then dy/dx = 1 / (dx/dy) and if you let y = f(x) and x = f^-1 (y), then dx/dy = 1 / (dy/dx). so derivatives of inverse functions can be written in 2 different ways? one of my problems asks to find the derivative of e^x (which i already know is e^x) by finding the derivative of the inverse of ln x. i did dy/dx = 1 / (1/y) = y. i also did dx/dy = 1 / (1/x) = x. how do i get e^x for either one? my equations are y = lnx and the inverse of that is x = lny.
• Nov 21st 2009, 07:52 PM
Em Yeu Anh
Hmm, well you had $x=lny$ and that $\frac{dy}{dx} = \frac{1}{\frac{1}{y}} = y$

From the original function you can see that $x = lny$ is equivalent to $e^x = y$ so I think you may have done it correctly. However my prof barely touched upon this subject so I'm sort of unfamiliar with it as well.
• Nov 22nd 2009, 03:51 AM
HallsofIvy
Quote:

Originally Posted by oblixps
this topic probably isn't supposed to be hard but i find it very confusing. my teacher went over it very briefly if not at all and my book doesn't help explain either. so it's my understanding that if you let y = f^-1 (x), then dy/dx = 1 / (dx/dy) and if you let y = f(x) and x = f^-1 (y), then dx/dy = 1 / (dy/dx). so derivatives of inverse functions can be written in 2 different ways? one of my problems asks to find the derivative of e^x (which i already know is e^x) by finding the derivative of the inverse of ln x. i did dy/dx = 1 / (1/y) = y. i also did dx/dy = 1 / (1/x) = x. how do i get e^x for either one? my equations are y = lnx and the inverse of that is x = lny.

You are getting your notation a bit confused. When talking about general functions, we might talk about f(x) and $f^{-1}(x)$ but when you have specific things like " $y= e^x$" it is best to write x= ln(y). From that, dx/dy= 1/y so [/tex]dy/dx= y= e^x[/tex]