Find the total differential for the function $w = 4 z ^ { 3 } y \sin x$
A) $d w = 4 z ^ { 3 } y \cos x d x - 4 z ^ { 3 } \cos x d y + 12 z ^ { 2 } y \sin x d z$
B) $d w = 4 z ^ { 3 } y \sin x d x - 4 z ^ { 3 } \sin x d y - 8 z ^ { 2 } y \sin x d z$
C) $d w = 4 z ^ { 3 } y \cos x d x + 8 z ^ { 3 } \sin x d y - 12 z ^ { 2 } y \sin x d z$
D) $d w = 8 z ^ { 3 } y \sin x d x + 4 z ^ { 3 } \sin x d y + 12 z ^ { 2 } y \sin x d z$
E) $d w = 4 z ^ { 3 } y \cos x d x + 4 z ^ { 3 } \sin x d y + 12 z ^ { 2 } y \sin x d z$

Question

Find the total differential for the function $w = 4 z ^ { 3 } y \sin x$ 
A) $d w = 4 z ^ { 3 } y \cos x d x - 4 z ^ { 3 } \cos x d y + 12 z ^ { 2 } y \sin x d z$
B) $d w = 4 z ^ { 3 } y \sin x d x - 4 z ^ { 3 } \sin x d y - 8 z ^ { 2 } y \sin x d z$
C) $d w = 4 z ^ { 3 } y \cos x d x + 8 z ^ { 3 } \sin x d y - 12 z ^ { 2 } y \sin x d z$
D) $d w = 8 z ^ { 3 } y \sin x d x + 4 z ^ { 3 } \sin x d y + 12 z ^ { 2 } y \sin x d z$
E) $d w = 4 z ^ { 3 } y \cos x d x + 4 z ^ { 3 } \sin x d y + 12 z ^ { 2 } y \sin x d z$

Quizplus · Accepted Answer

The total differential of a function $w = f(x, y, z)$ is given by $dw = \frac{\partial w}{\partial x}dx + \frac{\partial w}{\partial y}dy + \frac{\partial w}{\partial z}dz$. For $w = 4z^3y\sin x$, we calculate each partial derivative:- $\frac{\partial w}{\partial x} = 4z^3y\cos x$, so the term is $4z^3y\cos x dx$.- $\frac{\partial w}{\partial y} = 4z^3\sin x$, so the term is $4z^3\sin x dy$.- $\frac{\partial w}{\partial z} = 12z^2y\sin x$, so the term is $12z^2y\sin x dz$.Combining these gives the total differential as $dw = 4z^3y\cos x dx + 4z^3\sin x dy + 12z^2y\sin x dz$, which matches option E.

Find the Total Differential for the Function w=4z3ysin⁡xw = 4 z ^ { 3 } y \sin xw=4z3ysinx

Find the Total Differential for the Function $w = 4 z ^ { 3 } y \sin x$