Let $z = e ^ { 5 x y }$ . Then the differential $d ^ { z }$ is

A) $5 e ^ { 5 x y } ( y d x - x d y )$
B) $5 e ^ { 5 x y } ( y d x + x d y )$
C) $- 5 e ^ { 5 x y } ( y d x + x d y )$
D) $5 e ^ { 5 x y } ( - y d x + x d y )$
E) $e ^ { 5 x y } ( y d x + x d y )$

Question

Let $z = e ^ { 5 x y }$ . Then the differential $d ^ { z }$ is

A) $5 e ^ { 5 x y } ( y d x - x d y )$
B) $5 e ^ { 5 x y } ( y d x + x d y )$
C) $- 5 e ^ { 5 x y } ( y d x + x d y )$
D) $5 e ^ { 5 x y } ( - y d x + x d y )$
E) $e ^ { 5 x y } ( y d x + x d y )$

Quizplus · Accepted Answer

To find the differential $dz$ for $z = e^{5xy}$, we use the chain rule and product rule. Differentiating $z$ with respect to $x$ and $y$ gives $dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy$. The partial derivatives are $\frac{\partial z}{\partial x} = 5ye^{5xy}$ and $\frac{\partial z}{\partial y} = 5xe^{5xy}$, leading to $dz = 5e^{5xy}(ydx + xdy)$.

Let z=e5xyz = e ^ { 5 x y }z=e5xy Then the Differential

Let $z = e ^ { 5 x y }$ Then the Differential