To find the gradient of the function $f(x, y, z) = x e^{xy} + xy + z$, we need to compute the partial derivatives of $f$ with respect to each variable $x$, $y$, and $z$.

1. Partial derivative with respect to $x$:

We treat $y$ and $z$ as constants while differentiating with respect to $x$.

$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( x e^{xy} + xy + z \right)$ Using the product rule on $x e^{xy}$: $\frac{\partial}{\partial x} \left( x e^{xy} \right) = e^{xy} + x y e^{xy}$ Next, the derivative of $xy$ is just $y$, and the derivative of $z$ is 0 (since it's constant with respect to $x$).

Thus, $\frac{\partial f}{\partial x} = e^{xy} + x y e^{xy} + y$

2. Partial derivative with respect to $y$:

Now, treat $x$ and $z$ as constants and differentiate with respect to $y$:

$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( x e^{xy} + xy + z \right)$ Again, using the product rule on $x e^{xy}$: $\frac{\partial}{\partial y} \left( x e^{xy} \right) = x^2 e^{xy}$ The derivative of $xy$ with respect to $y$ is $x$, and the derivative of $z$ is 0.

Thus, $\frac{\partial f}{\partial y} = x^2 e^{xy} + x$

3. Partial derivative with respect to $z$:

For $z$, treat $x$ and $y$ as constants:

$\frac{\partial f}{\partial z} = \frac{\partial}{\partial z} \left( x e^{xy} + xy + z \right)$ Since $x e^{xy} + xy$ does not involve $z$, and the derivative of $z$ is 1, we have: $\frac{\partial f}{\partial z} = 1$

Final Gradient:

The gradient of the function is the vector of partial derivatives: $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$ Thus, $\nabla f = \left( e^{xy} + x y e^{xy} + y, x^2 e^{xy} + x, 1 \right)$

Would you like any more details or clarifications?

Here are 5 related questions to deepen your understanding:

1. What is the physical interpretation of the gradient in multivariable functions?
2. How does the product rule apply when differentiating the $x e^{xy}$ term?
3. How would the gradient change if the function had a different exponential term, like $x^2 e^{xy}$?
4. How can the gradient be used to find critical points of the function $f(x, y, z)$?
5. Can you compute the directional derivative of $f(x, y, z)$ in the direction of a given vector?

Tip: The gradient points in the direction of the steepest ascent of the function, and its magnitude gives the rate of the increase in that direction.