Find the partial derivative $\frac { \partial z } { \partial x } \text { for the function } z = \cos \left( x ^ { 8 } + y ^ { 8 } \right) \text {. }$
A) $\frac { \partial z } { \partial x } = - 8 x ^ { 7 } \sin \left( x ^ { 8 } + y ^ { 8 } \right)$
B) $\frac { \partial z } { \partial x } = 8 x ^ { 9 } \sin \left( x ^ { 9 } + y ^ { 9 } \right)$
C) $\frac { \partial z } { \partial x } = - 8 x ^ { 9 } \sin \left( x ^ { 8 } + y ^ { 8 } \right)$
D) $\frac { \partial z } { \partial x } = 8 x ^ { 9 } \cos \left( x ^ { 9 } + y ^ { 9 } \right)$
E) $\frac { \partial z } { \partial x } = 8 x ^ { 7 } \cos \left( x ^ { 8 } + y ^ { 8 } \right)$

Question

Find the partial derivative  $\frac { \partial z } { \partial x } \text { for the function } z = \cos \left( x ^ { 8 } + y ^ { 8 } \right) \text {. }$ 
A) $\frac { \partial z } { \partial x } = - 8 x ^ { 7 } \sin \left( x ^ { 8 } + y ^ { 8 } \right)$
B) $\frac { \partial z } { \partial x } = 8 x ^ { 9 } \sin \left( x ^ { 9 } + y ^ { 9 } \right)$
C) $\frac { \partial z } { \partial x } = - 8 x ^ { 9 } \sin \left( x ^ { 8 } + y ^ { 8 } \right)$
D) $\frac { \partial z } { \partial x } = 8 x ^ { 9 } \cos \left( x ^ { 9 } + y ^ { 9 } \right)$
E) $\frac { \partial z } { \partial x } = 8 x ^ { 7 } \cos \left( x ^ { 8 } + y ^ { 8 } \right)$

Quizplus · Accepted Answer

The correct partial derivative $\frac { \partial z } { \partial x }$ involves using the chain rule. Differentiating $\cos(x^8 + y^8)$ with respect to $x$ gives $-\sin(x^8 + y^8) \cdot 8x^7$, as the derivative of $x^8$ with respect to $x$ is $8x^7$, and the derivative of $\cos$ is $-\sin$.

Find the Partial Derivative ∂z∂x for the function z=cos⁡(x8+y8). \frac { \partial z } { \partial x } \text { for the function } z = \cos \left( x ^ { 8 } + y ^ { 8 } \right) \text {. }∂x∂z​ for the function z=cos(x8+y8).

Find the Partial Derivative $\frac { \partial z } { \partial x } \text { for the function } z = \cos \left( x ^ { 8 } + y ^ { 8 } \right) \text {. }$