Proving Zx + Zy = 0 using Chain Rule

[TranscendArcu]

Homework Statement

If Z= F(x-y), show that Z_x + Z_y = 0

Homework Equations

The Attempt at a Solution

Suppose I let Q = x-y. Then, by chain rule,

F_x(Q) * 1 + F_y(Q) * -1. By identity, this statement must hold for all values x,y. In particular, it must hold for x=y. By x=y,

F_y(Q) * 1 + F_y(Q) * -1 = 0.

Is this legitimate?

 

[murmillo]

I'm not sure what you mean by Fx(Q) * 1 + Fy(Q) * -1. Are you differentiating F(Q) with respect to x and to y?

 

[Deveno]

F is just a real-valued function, it only has one derivative.

if Q(x,y) = x-y, then Q has 2 partials, Q_x and Q_y.

using the chain rule we get:

Z_x = F'(Q)Q_x

your turn.