- #1

BucketOfFish

- 60

- 1

[tex]f(x,y)=xy[/tex]

where y is a function of x, say

[tex]y=x^2[/tex]

then the partial derivative of f with respect to x is

[tex]\frac{\partial f}{\partial x}=y[/tex]

However, if you substitute in y and express f as

[tex]f(x)=x^3[/tex]

then the partial derivative is

[tex]\frac{\partial f}{\partial x}=3x^2=3y[/tex]

despite the fact that these expressions for f are equivalent! What does it mean to change one variable (x) while holding another variable constant (y) even though it is a function of the first variable? Does this mean that if you choose different ways of expressing a system, you will end up with different partial derivatives? What influence does this have on solving such systems?