Provide an appropriate answer.
-Suppose that $w = x ^ { 2 } + y ^ { 2 } + 20 z + t$ and $x + 4 z + t = 8$. Assuming that the independent variables are $x$, $y$, and $z$, find $\frac { \partial w } { \partial x }$.
A) $\frac { \partial w } { \partial x } = 2 x - 5$
B) $\frac { \partial w } { \partial x } = 2 x + 5$
C) $\frac { \partial w } { \partial x } = 2 x + 1$
D) $\frac { \partial w } { \partial x } = 2 x - 1$

Question

Quizplus · Accepted Answer

To find $\frac{\partial w}{\partial x}$, we differentiate $w = x^2 + y^2 + 20z + t$ with respect to $x$, treating $y$ and $z$ as constants. This gives $\frac{\partial w}{\partial x} = 2x + \frac{\partial t}{\partial x}$. From $x + 4z + t = 8$, we differentiate both sides with respect to $x$ to find $\frac{\partial t}{\partial x} = -1$, since $\frac{\partial}{\partial x}(x + 4z + t) = 1 + 0 - \frac{\partial t}{\partial x} = 0$. Thus, $\frac{\partial w}{\partial x} = 2x - 1$.

Provide an Appropriate Answer w=x2+y2+20z+tw = x ^ { 2 } + y ^ { 2 } + 20 z + tw=x2+y2+20z+t

Provide an Appropriate Answer $w = x ^ { 2 } + y ^ { 2 } + 20 z + t$