Implicit Differentiation z=f(x/y) meaning

[xoxomae]

Mod note: Moved from the Homework section
1. Homework Statement
This might seem like a stupid question but I'm unsure what z= ƒ(x/y) means? I'm not sure how I would find ∂z/∂x , ∂z/∂y just from this statement either.

Thank you

Homework Equations

The Attempt at a Solution

 

[PeroK]

Homework Statement

This might seem like a stupid question but I'm unsure what z= ƒ(x/y) means? I'm not sure how I would find ∂z/∂x , ∂z/∂y just from this statement either.

Thank you

Homework Equations

The Attempt at a Solution

You could try taking $f$ to be some simple function. For example $f(X) = X^2$, where I've used $X$ to define the function to avoid confusion with $x, y$.

 

[Mark44]

Mod note: Moved from the Homework section
1. Homework Statement
This might seem like a stupid question but I'm unsure what z= ƒ(x/y) means? I'm not sure how I would find ∂z/∂x , ∂z/∂y just from this statement either.

It means that z is a function of the quotient x/y. For example, $z = (x/y)^2 + 3(x/y)$

To find the partial derivatives, you'll need to use the chain rule.

 

[xoxomae]

I'm sorry, I'm just really unsure how to apply the chain rule. So is ∂z/∂t = ∂/∂x(x/y) * F' ? But I'm unsure how to find all the other ones. I have the solutions attached to this post but I have no idea how to get them.
Thank You

 

[vela]

I'm sorry, I'm just really unsure how to apply the chain rule. So is ∂z/∂t = ∂/∂x(x/y) * F' ? But I'm unsure how to find all the other ones. I have the solutions attached to this post but I have no idea how to get them.
Thank You

What are $t$ and $F$?

Can you calculate $\frac{\partial z}{\partial x}$ if $z=f(u(x,y))$ where $u$ is a function of $x$ and $y$?

 

[xoxomae]

So i would let u=x/y so then

$$ \frac{\partial z}{\partial x} = \frac{\partial z }{\partial u} \frac{\partial u }{\partial x} = \frac{f'}{y} $$

cause I'm guessing $$\frac{\partial z }{\partial u} = f'$$ or am i completely wrong?

 

[vela]

That's right.