Find the total differential of the function $$z = 2 x ^ { 8 } y ^ { 3 }$$
A) $d z = 16 x ^ { 8 } y ^ { 3 } d x + 6 x ^ { 8 } y ^ { 3 } d y$
B) $d z = 48 x ^ { 7 } y ^ { 3 } d x + 48 x ^ { 8 } y ^ { 2 } d y$
C) $d z = 2 x ^ { 2 } y ^ { 8 } d x + 2 x ^ { 3 } y ^ { 7 } d y$
D) $d z = 16 x ^ { 7 } y ^ { 3 } d x + 6 x ^ { 8 } y ^ { 2 } d y$
E) $d z = 6 x ^ { 2 } y ^ { 8 } d x + 16 x ^ { 3 } y ^ { 7 } d y$

Question

Find the total differential of the function $$z = 2 x ^ { 8 } y ^ { 3 }$$ 
A) $d z = 16 x ^ { 8 } y ^ { 3 } d x + 6 x ^ { 8 } y ^ { 3 } d y$
B) $d z = 48 x ^ { 7 } y ^ { 3 } d x + 48 x ^ { 8 } y ^ { 2 } d y$
C) $d z = 2 x ^ { 2 } y ^ { 8 } d x + 2 x ^ { 3 } y ^ { 7 } d y$
D) $d z = 16 x ^ { 7 } y ^ { 3 } d x + 6 x ^ { 8 } y ^ { 2 } d y$
E) $d z = 6 x ^ { 2 } y ^ { 8 } d x + 16 x ^ { 3 } y ^ { 7 } d y$

Quizplus · Accepted Answer

Using the formula for total differential, we have 
$$d z = \frac { \partial z } { \partial x } d x + \frac { \partial z } { \partial y } d y$$
Taking partial derivatives of $z$ with respect to $x$ and $y$, we get
$$\frac{\partial z}{\partial x} = 16x^7y^3$$
$$\frac{\partial z}{\partial y} = 6x^8y^2$$
Substituting these values back into the equation for total differential, we get 
$$d z = 16 x ^ { 7 } y ^ { 3 } d x + 6 x ^ { 8 } y ^ { 2 } d y$$
Therefore, the correct answer is D.

Find the Total Differential of the Function z=2x8y3z = 2 x ^ { 8 } y ^ { 3 }z=2x8y3

Find the Total Differential of the Function $z = 2 x ^ { 8 } y ^ { 3 }$