# Implicit differentiation of one-parameter family

1. Jan 13, 2008

### bleucat

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

Use implicit differentiation to show that the one parameter family f(x, y)=c satisfies the differential equation dy/dx = $$-f_{x}/f_{y}$$, where $$f_{x}=\frac{\partial f}{\partial x}$$ and $$f_{y}=\frac{\partial f}{\partial y}$$.

2. Relevant equations

3. The attempt at a solution

Well, my teacher said I need to use the chain rule, but I'm confused about how to differentiate something that is in the general form f(x, y). And if f(x, y)=c, doesn't the derivative trivially equal 0?

Thanks in advance for the help.

2. Jan 13, 2008

### Vid

From the chain rule for the total derivative with respect to x:
$$\frac{\partial f}{\partial x}\frac{dx}{dx} +\frac{\partial f}{\partial y}\frac{dy}{dx} = 0$$

$$\frac{\partial f}{\partial x} +\frac{\partial f}{\partial y}\frac{dy}{dx} = 0$$

Solving for dy/dx gives -fx/fy.