1. ## Implicit Differentiation

$\displaystyle x^2+y^2+r^2-2s+13=0$

$\displaystyle x^3-y^3-r^3+3s+59=0$

How do I find the partial derivatives of x(r,s) or y(r,s) implicitly? I tried implicit differentiation and I got 2 different answers for either. Can someone show me any of the 4 derivatives step-by-step?

2. You can only find the values of the derivatives at specific points. Differentiate both for r, then both for s, and evaluate at $\displaystyle (r_0,s_0)$ to get:

$\displaystyle 2x(r_0,s_0)\frac{\partial x}{\partial r}(r_0,s_0)+2y(r_0,s_0)\frac{\partial y}{\partial r}(r_0,s_0)+2r_0=0$

$\displaystyle 3x^2(r_0,s_0)\frac{\partial x}{\partial r}(r_0,s_0)-3y^2(r_0,s_0)\frac{\partial y}{\partial r}(r_0,s_0)-3r_0^2=0$

$\displaystyle 2x(r_0,s_0)\frac{\partial x}{\partial s}(r_0,s_0)+2y(r_0,s_0)\frac{\partial y}{\partial s}(r_0,s_0)-2=0$

$\displaystyle 3x^2(r_0,s_0)\frac{\partial x}{\partial s}(r_0,s_0)-3y^2(r_0,s_0)\frac{\partial y}{\partial s}(r_0,s_0)+3=0$

The values $\displaystyle r_0, \ s_0, \ x(r_0,s_0), \ y(r_0,s_0)$ will be given, so the previous four equations form an algebraic system to be solved for $\displaystyle \frac{\partial x}{\partial s}(r_0,s_0), \ \frac{\partial y}{\partial s}(r_0,s_0)$.