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I am sorta skipping around at my own pace in Courant's Calculus book and came across partial derivatives. Are they geometrically the intersection of a plane and a surface? Why do we keep only one variable changing and the other variables fixed? Is it basically the definition of the derivative? For example

[tex] \frac{\partial y}{\partial x} [/tex] when [tex] f(x,y) = \sqrt{x^{2} + y^{2}} [/tex] Ok so would I consider y to be a constant when we want to find [tex] f_{x} [/tex] and vice versa for [tex] f_{y} [/tex]? Ok so this is what I did:

[tex] f_{x} = \frac{1}{2}\sqrt{x^2+y^2}2x [/tex]. But the answer is:

[tex] f_{x} = \frac{x}{\sqrt{x^2+y^2}} [/tex] and the same is true for [tex] f_{y} = \frac{y}{\sqrt{x^2+y^2}} [/tex] except the variables are reversed.

Any help is appreciated!

Thanks

[tex] \frac{\partial y}{\partial x} [/tex] when [tex] f(x,y) = \sqrt{x^{2} + y^{2}} [/tex] Ok so would I consider y to be a constant when we want to find [tex] f_{x} [/tex] and vice versa for [tex] f_{y} [/tex]? Ok so this is what I did:

[tex] f_{x} = \frac{1}{2}\sqrt{x^2+y^2}2x [/tex]. But the answer is:

[tex] f_{x} = \frac{x}{\sqrt{x^2+y^2}} [/tex] and the same is true for [tex] f_{y} = \frac{y}{\sqrt{x^2+y^2}} [/tex] except the variables are reversed.

Any help is appreciated!

Thanks

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