# Math Help - Implicit Function

1. ## Implicit Function

I have the following implicit equation as functions of x and y:

xu+yv+(u^2)-(v^2)-1=0
2xyuv-1=0

I found using Jacobian matrices etc. that the partial derivative of u with respect to x is:
-u(xu-yv+(2v^2))/(x(xu+(2u^2)-yv+(2v^2))

How do you find the partial derivative of this (ie the second partial derivative d^2u/dx^2, the second partial derivative of u with respect to x)?

Thank you for any help or clarification.

2. OK. Just answer this.

du/dx = partial derivative of u with respect to x =2u

Does d^(2)u/dx^2 = second partial derivative of u with respect to x = 2?

Is this right?

Can I just take the partial derivative of the already calculated partial derivative to find the second derivative?

3. Originally Posted by SwedishMan
OK. Just answer this.

du/dx = partial derivative of u with respect to x =2u

Does d^(2)u/dx^2 = second partial derivative of u with respect to x = 2?

Is this right?
No.
If $u=f(x,y)$
Them,
$(2u)_x=2u_x$
Not,
$(2u)_x=2$.
However, it CAN happen if $u=x+f(y)$

4. You cannot calculate the derivatives explicitly - or else the Implicitly function theorem would be pointless - only at specific points.

In this case, we get

$u_x+2uu_x-2v+u=0, \ y(uv+xu_xv+xuv_x)=0$

and use the conditions given -something like $u(x_0,y_0)=(a,b), v(x_0,y_0)=(c,d)$ - to solve algebraically this system for $u_x, v_x$. Same for the other derivatives.