# The derivative of a derivative

1. Mar 5, 2009

### Emethyst

1. The problem statement, all variables and given/known data
Using implicit differentation, find d^2y/dx^2 (the second derivative of y with respect to x) of the following in terms of x and y: (a) xy=4 (b)4y^2-3x^2=1

2. Relevant equations
All the simplifying laws for derivatives

3. The attempt at a solution
I found the derivatives for both the starting equations ((a) is -y/x and (b) is 3x/4y), but I cannot seem to find the derivatives of these derivatives. I know that the answer for (a) is 2y/x^2 and the answer for (b) is 3/16y^3, but I don't know where these answers come from. The way I have tried to solve for the second derivative always ends up with a dy/dx somewhere in the solution:

ex. for (a) I wound up with (dy/dx)(-x)+y/x^2 and for (b) 12y-(12x)(dy/dx)/16y^2

If someone could be of assistance for the second part of these questions and show me where i'm going wrong it would be greatly appreciated. Thanks in advance.

2. Mar 5, 2009

### Dick

Take the first one. Start with y'=(-y/x). Differentiate both sides using the quotient rule on the right. You'll get some y' terms on the right side, but you know y'=(-y/x).

3. Mar 5, 2009

### Emethyst

Thanks a lot for the help, can't believe I missed that :tongue: