Hoping my teacher's answer is wrong on partial derivatives..

[kankerfist]

I am studying my review for an upcoming test and one of the sample questions has an answer that I think (hope) is wrong. Can someone verify this and help me out if I am mistaken?

If w = x² – y² + z² and z = 4x² + y², find (∂w/∂x)_y

I am assuming this means find w_xy which I thought meant find the partial derivative of w with respect to x, then find the derivative of that result with respect to y. The teacher’s answer is:
2x + 2(4x² + y²)(8x)

My answer comes from the following:
First I find ∂w/∂x
w = x² – y² + (4x² + y²)²
So: ∂w/∂x = 2x + 2(4x² + y²)(8x)

My teacher stopped there, but I thought I now had to differentiate the above with respect to respect to y, which yields 32xy. Why did my teacher stop there?

 

[royhaas]

The notation is for the partial with respect to x, with y held constant. So your teacher stopped in the correct place. What you wanted to do would be labeled

\(\displaystyle \partial w /\partial x \partial y\).

 

[kankerfist]

Ok I must have my notation mixed up. So what if the above question were:
find dw/dx?

 

[pka]

The notation in your original post really puzzled me.
First, the expression \(\displaystyle \L
\left( {\frac{{\partial w}}{{\partial x}}} \right)_y\) is a mixed notation and not standard.
One usually sees \(\displaystyle \L
w_{x y}\) or its equivalent \(\displaystyle \L
\frac{{\partial ^2} w}{{\partial y\partial x}}\).
Secondly, I find it odd that the z variable is a function of x & y.

These make me wonder if the notation \(\displaystyle \L
\left( {\frac{{\partial w}}{{\partial x}}} \right)_y\) has a special meaning in the textbook of in the instructor’s notes. It would be helpful to know what textbook you are following.

If it is any help to you, had I seen that notation I would have assumed that it meant \(\displaystyle \L
w_{x y}\). But as I said, I have never seen that mixed notation.