Find the gradient of f with f(x,y) =xe^xy

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

### 1. Partial derivative with respect to \( x \):
We apply the product rule since \( f(x, y) = x \cdot e^{xy} \).

\[
\frac{\partial}{\partial x} [x \cdot e^{xy}] = \frac{\partial}{\partial x} x \cdot e^{xy} + x \cdot \frac{\partial}{\partial x} e^{xy}
\]

The derivative of \( x \) with respect to \( x \) is 1, and the derivative of \( e^{xy} \) with respect to \( x \) is:

\[
\frac{\partial}{\partial x} e^{xy} = y e^{xy}
\]

Thus, we have:

\[
\frac{\partial f}{\partial x} = e^{xy} + x \cdot y e^{xy} = e^{xy}(1 + xy)
\]

### 2. Partial derivative with respect to \( y \):
Now, we differentiate \( f(x, y) = x e^{xy} \) with respect to \( y \). Here, \( x \) is treated as a constant, and the derivative of \( e^{xy} \) with respect to \( y \) is:

\[
\frac{\partial}{\partial y} e^{xy} = x e^{xy}
\]

Thus:

\[
\frac{\partial f}{\partial y} = x^2 e^{xy}
\]

### Gradient of \( f(x, y) \):

The gradient of \( f(x, y) \) is the vector formed by the partial derivatives with respect to \( x \) and \( y \):

\[
\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = \left( e^{xy}(1 + xy), x^2 e^{xy} \right)
\]

Let me know if you need more details or have any questions!

Here are five related questions:
1. How would you find the critical points of \( f(x, y) = x e^{xy} \)?
2. What is the Hessian matrix for \( f(x, y) = x e^{xy} \)?
3. How does the gradient help in identifying the direction of steepest ascent?
4. What happens to the gradient when \( x = 0 \) or \( y = 0 \)?
5. How would you compute the directional derivative of \( f(x, y) \) in a given direction?

**Tip:** The gradient of a function at any point points in the direction of the steepest increase of the function at that point.

Find the gradient of f with f(x, y) = x e^xy.

Discover how to find the gradient of the function f(x, y) = x e^xy through detailed partial derivatives. This solution explores key concepts like the product rule and chain rule, making it ideal for students at the undergraduate level in calculus.

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