# Implicit Differentiation

1. Jul 9, 2009

### minderbinder

1. The problem statement, all variables and given/known data

Show that the set defined by the equations
$$x + y + z + w = sin(xyzw)$$

$$x - y + z + w^2 = cos(xyzw) - 1$$
can be described explicitly by equation of the form (z, w) = f(x, y) near the point (0,0,0,0); find the first partial derivatives of f(x,y) at the point (0,0)

2. Relevant equations

The above bolded part is the part I'm unsure about...

3. The attempt at a solution

I did:
$$G = x + y + z + w - sin(xyzw)$$

$$H = x - y + z + w^2 - cos(xyzw) + 1$$

$$\frac{\partial (G, H)}{\partial (x, y)} + \frac{\partial (G, H)}{\partial (z, w)} \frac{\partial f}{\partial x } = 0$$

Then I solved for $$\frac{\partial f}{\partial x}$$?

2. Jul 9, 2009

### Fenn

I think the note of "near the point (0,0,0,0)" is a clue that you can use the small angle approximation of

$$\sin\theta = \theta,\quad \cos\theta = 1,\quad \texttt{when }\theta \texttt{ is small.}$$

I'm not fully understanding what the question is asking, but I'm interpreting it as saying you need to find

$$z = f_1(x,y)$$

$$w = f_2(x,y)$$

From there, I would calculate
$$\left.\frac{\partial z}{\partial x}\right|_{(0,0)} = \frac{\partial f_1(0,0)}{\partial x}$$
$$\left.\frac{\partial z}{\partial y}\right|_{(0,0)} = \frac{\partial f_1(0,0)}{\partial y}$$
$$\left.\frac{\partial w}{\partial x}\right|_{(0,0)} = \frac{\partial f_2(0,0)}{\partial x}$$
$$\left.\frac{\partial w}{\partial y}\right|_{(0,0)} = \frac{\partial f_2(0,0)}{\partial y}$$

As I said, though, I may not be properly understanding the question.