Find the value of the gradient vector field of the function $f ( x , y ) = e ^ { xy ^ { 2 } }$ at the point $( 1,1 )$ .
A) $e \mathbf { i}$
B)$e \mathrm { j }$
C)$2 e \mathbf { i }$
D)$2 e \mathrm { j }$
E)$e \mathbf { i } + e \mathbf { j }$
F)$e \mathbf { i } + 2 e \mathbf { j }$
G) $2 e \mathbf { i } + 2 e \mathbf { j }$
H)$2 e \mathbf { i } + e \mathbf { j }$

Question

Find the value of the gradient vector field of the function $f ( x , y ) = e ^ { xy ^ { 2 } }$ at the point $( 1,1 )$ .
A) $e \mathbf { i}$ 
B)$e \mathrm { j }$ 
C)$2 e \mathbf { i }$
D)$2 e \mathrm { j }$ 
E)$e \mathbf { i } + e \mathbf { j }$ 
F)$e \mathbf { i } + 2 e \mathbf { j }$ 
G) $2 e \mathbf { i } + 2 e \mathbf { j }$ 
H)$2 e \mathbf { i } + e \mathbf { j }$

Quizplus · Accepted Answer

The gradient vector field of $f(x, y) = e^{xy^2}$ is computed by taking the partial derivatives with respect to $x$ and $y$, respectively. The partial derivative with respect to $x$ is $y^2e^{xy^2}$, and with respect to $y$ is $2xye^{xy^2}$. Evaluating these at the point $(1,1)$ gives $e$ for the $x$-component and $2e$ for the $y$-component, resulting in the vector $e\mathbf{i} + 2e\mathbf{j}$.

Find the Value of the Gradient Vector Field of the Function