Given the function below, determine the gradient of the function.
f (x, y, z) = x 𝑒𝑥𝑦+ xy+z

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.

Given the function below, determine the gradient of the function. f(x, y, z) = x e^{xy} + xy + z

Learn how to compute the gradient of the function f(x, y, z) = x e^{xy} + xy + z by finding the partial derivatives with respect to x, y, and z. This problem covers key multivariable calculus concepts and involves applying the product and chain rules. Suitable for students in advanced calculus.

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