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## Homework Statement

The equation ## f(x,y) = f(a,b) ## defines a level curve through a point ## (a,b) ## where ## \nabla f(a,b) \neq \vec 0##. Use implicit differentiation and the chain rule to show that the slope of the line tangent to this curve at the point ##(a,b)## is ##-f_x(a,b)/f_y(a,b)## if ##f_y \neq 0##.

## Homework Equations

##\nabla f(x,y) = f_x(x,y)\vec i + f_y(x,y)\vec j##

## The Attempt at a Solution

Well, if we're using implicit differentiation, then I think ##y = f(x)## and ##\frac{\partial f(x,y)}{\partial y} = \frac{df(x,f(x))}{dx}\frac{dx}{dx}##

That would make it: ##\frac{df(x,y)}{dx} = \frac{df(x)}{dx}\frac{dx}{dx} + \frac{df((f(x))}{dx}\frac{dx}{dx}##

Since dx/dx = 1, that becomes: ##\frac{df(x,y)}{dx} = \frac{df(x)}{dx} + \frac{df((f(x))}{dx}##

However, the right term appears to go to zero (or right "branch" as my math teacher told us), leaving this as: ##\frac{df(x,y)}{dx} = \frac{df(x)}{dx}##

Assuming all that's correct, which I'm not sure it is, I'm now stuck. I can't see how to get the slope equal to ##-f_x(a,b)/f_y(a,b)##