# Compute these partial derivatives with u = u(x,y), v = v(x,y); (x,y) = (1,1) = (u,v).

## Homework Statement

The equations xu^2 + yv = 2, 2yv^2 + xu = 3 define u(x,y) and v(x,y) in terms of x and y near the point (x,y) = (1,1) and (u,v) = (1,1).

Compute the following partial derivatives:
(A) ∂u/∂x(1,1)
(B) ∂u/∂y(1,1)
(C) ∂v/∂x(1,1)
(D) ∂v/∂y(1,1)

The answers are:
(A) ∂u/∂x(1,1) = -0.428571428571429
(B) ∂u/∂y(1,1) = -0.285714285714286
(C) ∂v/∂x(1,1) = -0.142857142857143
(D) ∂v/∂y(1,1) = -0.428571428571429

## Homework Equations

To my knowledge: partial differentiation and implicit differentiation.

## The Attempt at a Solution

I tried implicitly and partially differentiating xu^2 + yv = 2 and got:
u^2 + 2xu∂u/∂x = 0
∂u/∂x = -u^2 /(2xu)
∂u/∂x(1,1) = -(1)^2/(2*1*1) = -1/2 (which is close to the answer but not good enough).

Could someone please tell me what I am doing wrong and how to do this correctly?

Any help would be greatly appreciated!
Thanks in advance!

## Answers and Replies

Dick
Science Advisor
Homework Helper

Where is the ∂v/∂x term in your partial derivative? You'll need to differentiate both equations with respect to both x and y and then treat it as a system of linear equations in the four unknowns.

Your advice worked for me. Thanks!