https://file.aiquickdraw.com/mathbot/file/1726241587915jo5d0dxl.jpg

The mathematical expression in the image appears to involve a partial derivative:

\[
\frac{\partial}{\partial y} \left( x^2 \frac{\partial^2 p}{\partial x^2} \right)
\]

This expression indicates that we are taking the partial derivative with respect to \( y \) of the entire term inside the parentheses. The term inside is a product of \( x^2 \) and a second-order partial derivative of \( p \) with respect to \( x \).

### Breaking this down:

1. The outer derivative is with respect to \( y \), but the inside terms are independent of \( y \), so this could mean the derivative with respect to \( y \) would directly yield zero unless some relation with \( y \) is hidden.
   
2. The term inside involves:
   \[
   x^2 \frac{\partial^2 p}{\partial x^2}
   \]
   which is the product of \( x^2 \) and the second partial derivative of \( p \) with respect to \( x \).

Would you like me to go further into specific derivative rules or clarify more on any particular step?

### 5 Related Questions:
1. What are the rules for taking partial derivatives of products like \( x^2 \frac{\partial^2 p}{\partial x^2} \)?
2. How would this expression change if \( p \) also depended on \( y \)?
3. What are the physical or practical applications of such second-order partial differential equations?
4. Could this expression be part of a larger equation, such as one in fluid dynamics or thermodynamics?
5. How does the order of differentiation affect the outcome when differentiating with respect to multiple variables?

**Tip:** When working with mixed partial derivatives, always check if the function dependencies on variables like \( x \) and \( y \) are explicit. This helps simplify expressions or identify when terms vanish.

What is the partial derivative expression \( \frac{\partial}{\partial y} \left( x^2 \frac{\partial^2 p}{\partial x^2} \right) \)?

Learn the steps to compute the partial derivative \( \frac{\partial}{\partial y} \left( x^2 \frac{\partial^2 p}{\partial x^2} \right) \), breaking down the product rule and understanding second-order partial derivatives. This guide is ideal for undergraduate students studying calculus and differential equations.

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