Find $\frac { \partial f } { \partial z }$ for $f ( x , y , z ) = e ^ { x y ^ { 2 } z ^ { 3 } }$
A) $y ^ { 2 } z ^ { 3 } e ^ { x y ^ { 2 } z ^ { 3 } }$
B) $2 x y z ^ { 3 } e ^ { x y ^ { 2 } z ^ { 3 } }$
C) $3 x y ^ { 2 } z ^ { 2 } e ^ { x y ^ { 2 } z ^ { 3 } }$
D) $x y ^ { 2 } z ^ { 3 } e ^ { x y ^ { 2 } z ^ { 3 } }$

Question

Quizplus · Accepted Answer

To find $\frac { \partial f } { \partial z }$, we apply the chain rule. Differentiating $e ^ { x y ^ { 2 } z ^ { 3 } }$ with respect to $z$, we treat $x$ and $y$ as constants. The derivative of $e^{u}$ with respect to $u$ is $e^{u}$, and the derivative of $u = x y ^ { 2 } z ^ { 3 }$ with respect to $z$ is $3 x y ^ { 2 } z ^ { 2 }$. Thus, $\frac { \partial f } { \partial z } = 3 x y ^ { 2 } z ^ { 2 } e ^ { x y ^ { 2 } z ^ { 3 } }$.

Find ∂f∂z\frac { \partial f } { \partial z }∂z∂f​ For f(x,y,z)=exy2z3f ( x , y , z ) = e ^ { x y ^ { 2 } z ^ { 3 } }f(x,y,z)=exy2z3

Find $\frac { \partial f } { \partial z }$ For $f ( x , y , z ) = e ^ { x y ^ { 2 } z ^ { 3 } }$