Find the partial derivative $\frac { \partial z } { \partial y } \text { for the function } z = \cos \left( x ^ { 5 } + y ^ { 5 } \right)$
A) $\frac { \partial z } { \partial y } = - 5 y ^ { 4 } \sin \left( x ^ { 5 } + y ^ { 5 } \right)$
B) $\frac { \partial z } { \partial y } = - 5 y ^ { 6 } \sin \left( x ^ { 5 } + y ^ { 5 } \right)$
C) $\frac { \partial z } { \partial y } = 5 y ^ { 6 } \sin \left( x ^ { 6 } + y ^ { 6 } \right)$
D) $\frac { \partial z } { \partial y } = 5 y ^ { 6 } \cos \left( x ^ { 6 } + y ^ { 6 } \right)$
E) $\frac { \partial z } { \partial y } = 5 y ^ { 4 } \cos \left( x ^ { 5 } + y ^ { 5 } \right)$

Question

Quizplus · Accepted Answer

We take the partial derivative of z with respect to y as follows:
\begin{align*}
\frac{\partial z}{\partial y} &= \frac{\partial}{\partial y}[\cos(x^5+y^5)]\
&= -\sin(x^5+y^5)\cdot \frac{\partial}{\partial y}[x^5+y^5]\
&= -\sin(x^5+y^5)\cdot [5y^4]\
&= -5y^4\sin(x^5+y^5)
\end{align*}
Therefore, the correct choice is $\boxed{	extbf{(A)}}$.

Find the Partial Derivative ∂z∂y for the function z=cos⁡(x5+y5)\frac { \partial z } { \partial y } \text { for the function } z = \cos \left( x ^ { 5 } + y ^ { 5 } \right)∂y∂z​ for the function z=cos(x5+y5)

Find the Partial Derivative $\frac { \partial z } { \partial y } \text { for the function } z = \cos \left( x ^ { 5 } + y ^ { 5 } \right)$