To find the partial derivatives of the given function:

$z = x \sin(xy)$

we'll differentiate $z$ with respect to $x$ and $y$.

1. Partial derivative of $z$ with respect to $x$:

We treat $y$ as a constant while differentiating with respect to $x$.

$\frac{\partial z}{\partial x} = \frac{\partial}{\partial x} \left( x \sin(xy) \right)$

Using the product rule:

$\frac{\partial z}{\partial x} = \sin(xy) + x \cdot \frac{\partial}{\partial x} \left( \sin(xy) \right)$

Now, applying the chain rule to differentiate $\sin(xy)$ with respect to $x$:

$\frac{\partial}{\partial x} \left( \sin(xy) \right) = \cos(xy) \cdot y$

Therefore, the partial derivative with respect to $x$ is:

$\frac{\partial z}{\partial x} = \sin(xy) + x \cos(xy) \cdot y$

2. Partial derivative of $z$ with respect to $y$:

Now, treat $x$ as a constant while differentiating with respect to $y$:

$\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} \left( x \sin(xy) \right)$

Using the chain rule:

$\frac{\partial z}{\partial y} = x \cdot \cos(xy) \cdot x$

This simplifies to:

$\frac{\partial z}{\partial y} = x^2 \cos(xy)$

Summary of Results:

• $\frac{\partial z}{\partial x} = \sin(xy) + x \cdot y \cos(xy)$
• $\frac{\partial z}{\partial y} = x^2 \cos(xy)$

Let me know if you want more details or further explanations.

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Here are some related questions for you to explore:

1. What is the second partial derivative of $z$ with respect to $x$?
2. How would you calculate the mixed partial derivative $\frac{\partial^2 z}{\partial x \partial y}$?
3. What would happen if you took the derivative of $z$ with respect to time, assuming $x$ and $y$ are both functions of time?
4. Can you generalize the method used for partial derivatives to higher-order functions of multiple variables?
5. How do partial derivatives relate to the concept of tangent planes in multivariable calculus?

Tip: When using the chain rule in partial derivatives, always carefully distinguish between the variable you're differentiating with respect to and the others that are treated as constants.