To compute the partial derivatives $f_x$ and $f_y$ of the function

$f(x, y) = x^2 e^{3xy},$

we will differentiate $f(x, y)$ with respect to $x$ and $y$, treating the other variable as a constant.

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1. Partial derivative with respect to $x$ ($f_x$):

The product rule is required because $f(x, y)$ is a product of $x^2$ and $e^{3xy}$. The product rule states:

$\frac{\partial}{\partial x}(u v) = u'v + uv',$

where $u = x^2$ and $v = e^{3xy}$.

Step 1: Differentiate $u = x^2$ with respect to $x$:

$\frac{\partial}{\partial x}(x^2) = 2x.$

Step 2: Differentiate $v = e^{3xy}$ with respect to $x$:

Using the chain rule: $\frac{\partial}{\partial x}(e^{3xy}) = e^{3xy} \cdot \frac{\partial}{\partial x}(3xy) = e^{3xy} \cdot 3y.$

Step 3: Apply the product rule:

$f_x = \frac{\partial}{\partial x}(x^2 e^{3xy}) = (2x)e^{3xy} + (x^2)(3y e^{3xy}).$

Simplify: $f_x = 2x e^{3xy} + 3x^2 y e^{3xy}.$

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2. Partial derivative with respect to $y$ ($f_y$):

We use the product rule again, where $u = x^2$ and $v = e^{3xy}$.

Step 1: Differentiate $u = x^2$ with respect to $y$:

Since $x^2$ does not depend on $y$: $\frac{\partial}{\partial y}(x^2) = 0.$

Step 2: Differentiate $v = e^{3xy}$ with respect to $y$:

Using the chain rule: $\frac{\partial}{\partial y}(e^{3xy}) = e^{3xy} \cdot \frac{\partial}{\partial y}(3xy) = e^{3xy} \cdot 3x.$

Step 3: Apply the product rule:

$f_y = \frac{\partial}{\partial y}(x^2 e^{3xy}) = (0)e^{3xy} + (x^2)(3x e^{3xy}).$

Simplify: $f_y = 3x^3 e^{3xy}.$

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Final Results:

$f_x = 2x e^{3xy} + 3x^2 y e^{3xy},$ $f_y = 3x^3 e^{3xy}.$

Let me know if you’d like a step-by-step explanation of any part of this!

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Questions to Expand:

1. How does the chain rule apply when differentiating exponentials?
2. What is the significance of partial derivatives in multivariable functions?
3. Can we apply these results to evaluate $f_x$ and $f_y$ at specific points (e.g., $x=1, y=0$)?
4. How do we visualize partial derivatives geometrically?
5. What are higher-order partial derivatives for $f(x, y)$?

Tip:

Always keep track of which variable is being treated as constant when taking partial derivatives!