Partial Differentiation Confusion

[RazerM]

Homework Statement

Find $$\frac{\partial z}{\partial x} \frac{\partial z}{\partial y}$$ where $$z=\left( [x+y]^3-4y^2 \right)^{\frac{1}{2}}$$

Homework Equations

-

The Attempt at a Solution

I know that $$\frac{\partial z}{\partial y}=\frac{3(x+y)^2-8y}{2\sqrt{(x+y)^3-4y^2}}$$
but I am unsure whether $$\frac{\partial z}{\partial x}$$ is the exact same or does not include the '-8y' in the numerator.

I get the feeling that when finding the derivative inside the square root (in z) that y should still be treated as constant and therefore have no -8y.

 

[tiny-tim]

Welcome to PF!

Hi RazerM! Welcome to PF!

(on this forum, you need to type "tex", not "TEX" )

Homework Statement

Find $$\frac{\partial z}{\partial x} \frac{\partial z}{\partial y}$$ where $$z=\left( [x+y]^3-4y^2 \right)^{\frac{1}{2}}$$

Homework Equations

-

The Attempt at a Solution

I know that $$\frac{\partial z}{\partial y}=\frac{3(x+y)^2-8y}{2\sqrt{(x+y)^3-4y^2}}$$
but I am unsure whether $$\frac{\partial z}{\partial x}$$ is the exact same or does not include the '-8y' in the numerator.

I get the feeling that when finding the derivative inside the square root (in z) that y should still be treated as constant and therefore have no -8y.

Yes, that's completely correct.

∂z/∂x means "keeping y constant", so that's exactly what you do!

 

[RazerM]

So that means $$\frac{\partial z}{\partial x}=\frac{3(x+y)^2}{2\sqrt{(x+y)^3-4y^2}}$$?

 

[tiny-tim]

Yup!

(nice LaTeX, btw )

 

[RazerM]

Thanks :)

I taught myself to use LaTeX to help me with my Physics Investigation as part of Advanced Higher Physics (Highest level of physics taught in school - Scotland), we never got told to use it but no way was I using MS Office or Openoffice's limited equation typesetting, would have been a nightmare :P

 

[tiny-tim]

One of the many benefits of PF membership is that you can now use LaTeX as much as you like!

(in case you haven't found anything similar, a useful bookmark is http://www.physics.udel.edu/~dubois/lshort2e/node61.html#SECTION008100000000000000000" )