if y e ^xy find double order partial derivative with respect to x

To find the second-order partial derivative of the function \( y e^{xy} \) with respect to \( x \), we follow these steps:

### 1. First-order partial derivative with respect to \( x \)

The given function is:
\[
f(x, y) = y e^{xy}
\]

Differentiate this with respect to \( x \), treating \( y \) as a constant:
\[
\frac{\partial}{\partial x} \left( y e^{xy} \right)
\]

Use the product rule, where \( u(x) = y \) and \( v(x) = e^{xy} \):
\[
\frac{\partial}{\partial x} \left( y e^{xy} \right) = y \frac{\partial}{\partial x} \left( e^{xy} \right)
\]

Now, differentiate \( e^{xy} \) with respect to \( x \):
\[
\frac{\partial}{\partial x} \left( e^{xy} \right) = y e^{xy}
\]

So the first-order partial derivative becomes:
\[
\frac{\partial}{\partial x} \left( y e^{xy} \right) = y^2 e^{xy}
\]

### 2. Second-order partial derivative with respect to \( x \)

Now, take the derivative of \( y^2 e^{xy} \) with respect to \( x \):
\[
\frac{\partial}{\partial x} \left( y^2 e^{xy} \right)
\]

Again, use the product rule, where \( u(x) = y^2 \) and \( v(x) = e^{xy} \):
\[
\frac{\partial}{\partial x} \left( y^2 e^{xy} \right) = y^2 \frac{\partial}{\partial x} \left( e^{xy} \right)
\]

Since \( \frac{\partial}{\partial x} \left( e^{xy} \right) = y e^{xy} \), we get:
\[
\frac{\partial^2}{\partial x^2} \left( y e^{xy} \right) = y^3 e^{xy}
\]

Thus, the second-order partial derivative of \( y e^{xy} \) with respect to \( x \) is:
\[
\frac{\partial^2}{\partial x^2} \left( y e^{xy} \right) = y^3 e^{xy}
\]

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Do you want further details or have any questions? Here are some related questions for deeper understanding:

1. What is the first-order partial derivative of \( e^{xy} \) with respect to \( y \)?
2. How do you find mixed partial derivatives of a function involving exponential terms?
3. Can you generalize the process for finding higher-order derivatives of \( e^{xy} \)?
4. What is the second-order mixed partial derivative of \( y e^{xy} \) with respect to \( x \) and \( y \)?
5. How does the product rule apply when differentiating exponential functions?

**Tip**: When differentiating exponential functions like \( e^{xy} \), always remember to use the chain rule for any non-trivial exponent terms.

This page provides a step-by-step guide to finding the second-order partial derivative of the function y e^{xy} with respect to x. Learn about key calculus concepts like the product rule and chain rule, essential for handling exponential functions. Suitable for college-level calculus students.

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