Function Composition Homework: Derivatives of z w.r.t x, y

[Ikastun]

Homework Statement

Be z=F(u,v,w), u=f(x,y), v=e^-αx, w=ln y, get the expression $\partial$z/$\partial$x, $\partial$z/$\partial$y.

Homework Equations

Chain rule.

The Attempt at a Solution

$\partial$z/$\partial$x=$\partial$z/$\partial$v*dv/dx=-α e^-αx

$\partial$z/$\partial$y=$\partial$z/$\partial$w*dw/dy=1/y

 

[ShayanJ]

When calculating $\frac{\partial z}{\partial x}$,you should consider all variables that depend on x not only v!

 

[Ikastun]

Thank you. What about the following expression: ∂z/∂x=∂z/∂u*∂u/∂x+∂z/∂v*dv/dx?. Same to ∂z/∂y with w.

 

[Mark44]

Thank you. What about the following expression: ∂z/∂x=∂z/∂u*∂u/∂x+∂z/∂v*dv/dx?.

Looks good. You even picked up on the fact that dv/dx is a regular (not partial) derivative.

Same to ∂z/∂y with w.

What did you get?

 

[Ikastun]

∂z/∂y=∂z/∂u*∂u/∂y+∂z/∂w*dw/dy.

 

[Mark44]

Looks good!