To determine the gradient vector of the function $f(x, y) = e^x (x \sin(2y) + y \cos(2y))$, we need to find the partial derivatives of $f$ with respect to $x$ and $y$.

Step 1: Partial Derivative with Respect to $x$

The partial derivative of $f$ with respect to $x$, $f_x$, is calculated as follows:

1. Apply the product rule, where:
   • The first term is $e^x$,
   • The second term is $x \sin(2y) + y \cos(2y)$.

$f_x = \frac{\partial}{\partial x} \left( e^x (x \sin(2y) + y \cos(2y)) \right)$

2. By the product rule, this becomes: $f_x = e^x \cdot \frac{\partial}{\partial x}(x \sin(2y) + y \cos(2y)) + (x \sin(2y) + y \cos(2y)) \cdot \frac{\partial}{\partial x}(e^x)$

3. Simplify each part:

   • The derivative of $e^x$ with respect to $x$ is $e^x$.
   • The derivative of $x \sin(2y) + y \cos(2y)$ with respect to $x$ is $\sin(2y)$ (since $y$ is treated as a constant with respect to $x$).

So: $f_x = e^x (\sin(2y)) + (x \sin(2y) + y \cos(2y)) e^x$

4. Combine terms: $f_x = e^x (\sin(2y) + x \sin(2y) + y \cos(2y))$

Step 2: Partial Derivative with Respect to $y$

Now, let's find $f_y$, the partial derivative of $f$ with respect to $y$.

$f_y = \frac{\partial}{\partial y} \left( e^x (x \sin(2y) + y \cos(2y)) \right)$

Since $e^x$ is a constant with respect to $y$, we can factor it out:

$f_y = e^x \cdot \frac{\partial}{\partial y}(x \sin(2y) + y \cos(2y))$

1. Differentiate $x \sin(2y) + y \cos(2y)$ with respect to $y$:

   • The derivative of $x \sin(2y)$ with respect to $y$ is $2x \cos(2y)$,
   • The derivative of $y \cos(2y)$ with respect to $y$ is $\cos(2y) - 2y \sin(2y)$.

2. So we have: $f_y = e^x (2x \cos(2y) + \cos(2y) - 2y \sin(2y))$

Final Gradient Vector

The gradient vector $\nabla f(x, y)$ is:

$\nabla f(x, y) = \left( f_x, f_y \right)$

Substitute in the expressions for $f_x$ and $f_y$:

$\nabla f(x, y) = \left( e^x (\sin(2y) + x \sin(2y) + y \cos(2y)), \, e^x (2x \cos(2y) + \cos(2y) - 2y \sin(2y)) \right)$

Would you like more details on any of these steps, or help with another aspect of this problem?

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Here are 5 related questions you might find helpful:

1. What is the significance of the gradient vector in calculus?
2. How does one interpret the direction and magnitude of the gradient vector?
3. Could you find the critical points of $f(x, y)$ using this gradient?
4. How would you determine if $f(x, y)$ has any local maxima or minima?
5. Can this method of finding the gradient be applied to higher-dimensional functions?

Tip: The gradient vector always points in the direction of the steepest increase of a function at a given point.