Understanding Partial Derivatives: Solving for f'(x) at a Specific Point

[theBEAST]

Homework Statement

If f(x,y) = x(x^2+y^2)^(-3/2)*e^(sin(x^2y)) find the derivative of f with respect to x at the point (1,0).

The Attempt at a Solution

The textbook solution just plugs 0 into y and gets f(x) = x^-2 and then proceeds to differentiate this resulting in the answer -2. I don't understand why this is legal. How can you just plug the point into the function and then take the derivative?

For example, if I had the function y = x and I wanted the derivative at x=0. You can't just plug in zero and take the derivative...?

 

[Dick]

Because when you taking a partial derivative of a function of two variables f(x,y) with respect to x you assume y is a constant. You can either find it for any y or plug in a value for y to being with. y=x isn't a function of two variables.