https://file.aiquickdraw.com/mathbot/file/17266629745174xssgrzw.jpg

The function given in the image is:

\[
f(x, y) = 3x^3 - 2x^2y + 5xy^2 + 7y^3 + 14
\]

You are asked to evaluate the second-order mixed partial derivative:

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

### Steps to solve:

1. **First, find** \( \frac{\partial f}{\partial x} \) (the first partial derivative of \( f \) with respect to \( x \)).

\[
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( 3x^3 - 2x^2y + 5xy^2 + 7y^3 + 14 \right)
\]

\[
= 9x^2 - 4xy + 5y^2
\]
(Since terms involving only \( y \) or constants disappear.)

2. **Next, find** \( \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) \).

\[
\frac{\partial}{\partial y} (9x^2 - 4xy + 5y^2) = -4x + 10y
\]

Thus, the second-order mixed partial derivative is:

\[
\frac{\partial^2 f}{\partial x \partial y} = -4x + 10y
\]

Would you like any further explanation or details on these steps?

Here are some related questions to extend your understanding:
1. How do you compute a second-order mixed partial derivative for a more complex function?
2. What does the symmetry of mixed partial derivatives (\( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \)) imply?
3. How would the partial derivatives change if \( f(x, y) \) had an exponential term?
4. Can second-order partial derivatives provide information about the curvature of surfaces?
5. How can we apply mixed partial derivatives to find critical points in multivariable calculus?

**Tip:** Always check if the function is continuously differentiable to apply Clairaut's theorem for mixed partial derivatives symmetry.

Evaluate the second-order mixed partial derivative of the function f(x, y) = 3x^3 - 2x^2y + 5xy^2 + 7y^3 + 14.

Learn how to evaluate the second-order mixed partial derivative of the multivariable function f(x, y) = 3x^3 - 2x^2y + 5xy^2 + 7y^3 + 14. The solution involves computing partial derivatives and applying multivariable calculus concepts.

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