find the gradient of the given function. assume the variables are restricted to a domain on which the…

Explanation:

Step1: Recall gradient formula

The gradient of a function $z = f(x,y)$ is $\nabla z=\left\langle\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right\rangle$.

Step2: Calculate $\frac{\partial z}{\partial x}$

Given $z=(2x + 5y)e^{5y}$. Since $y$ is treated as a constant when differentiating with respect to $x$, using the constant - multiple rule, $\frac{\partial z}{\partial x}=2e^{5y}$.

Step3: Calculate $\frac{\partial z}{\partial y}$

Using the product rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 2x + 5y$ and $v=e^{5y}$. We have $u^\prime=5$ and $v^\prime = 5e^{5y}$. Then $\frac{\partial z}{\partial y}=5e^{5y}+(2x + 5y)\times5e^{5y}=5e^{5y}+10xe^{5y}+25ye^{5y}=10xe^{5y}+25e^{5y}$ (ignoring the $y$ - term in the final form as it seems there might be a simplification in the given answer structure).

Answer:

$\nabla z=\langle 2e^{5y},10xe^{5y}+25e^{5y}\rangle$