Find the requested partial derivative.
-$$( \partial w / \partial z ) _ { x , y } \text { at } ( x , y , z , w ) = ( 1,2,9,266 ) \text { if } w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 10 x y z$$
A) 28
B) 33
C) 38
D) 48

Question

Quizplus · Accepted Answer

The partial derivative of $w$ with respect to $z$, holding $x$ and $y$ constant, is given by the derivative of $w = x^2 + y^2 + z^2 + 10xyz$ with respect to $z$. This derivative is $\frac{\partial w}{\partial z} = 2z + 10xy$. Substituting $x = 1$, $y = 2$, and $z = 9$ into this expression gives $2(9) + 10(1)(2) = 18 + 20 = 38$.