https://file.aiquickdraw.com/mathbot/file/1727470370118bsxqhf3c.jpg

The expression in the image is:

\[
\frac{\partial}{\partial y} \left( x^2 \tan^{-1} \left( \frac{y}{x} \right) \right)
\]

To solve this partial derivative, we'll apply the following steps:

### Step 1: Identify the structure
The function consists of \( x^2 \) multiplied by the inverse tangent function ( \( \tan^{-1} \left( \frac{y}{x} \right) \)).

Since \( x^2 \) is independent of \( y \), we can treat it as a constant with respect to \( y \), and the derivative only needs to be applied to \( \tan^{-1} \left( \frac{y}{x} \right) \).

### Step 2: Differentiate the arctangent function
The derivative of \( \tan^{-1}(z) \) with respect to \( z \) is given by:
\[
\frac{d}{dz} \left( \tan^{-1}(z) \right) = \frac{1}{1 + z^2}
\]
Here, \( z = \frac{y}{x} \), and we will apply the chain rule to differentiate \( \tan^{-1} \left( \frac{y}{x} \right) \) with respect to \( y \).

\[
\frac{\partial}{\partial y} \left( \tan^{-1} \left( \frac{y}{x} \right) \right) = \frac{1}{1 + \left( \frac{y}{x} \right)^2} \cdot \frac{1}{x}
\]

Simplifying:

\[
\frac{\partial}{\partial y} \left( \tan^{-1} \left( \frac{y}{x} \right) \right) = \frac{1}{x^2 + y^2}
\]

### Step 3: Multiply by \( x^2 \)
Now, multiply the result by \( x^2 \), which we treated as constant earlier:

\[
\frac{\partial}{\partial y} \left( x^2 \tan^{-1} \left( \frac{y}{x} \right) \right) = x^2 \cdot \frac{1}{x^2 + y^2}
\]

### Final Answer:
\[
\frac{x^2}{x^2 + y^2}
\]

Let me know if you need further details or explanations!

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### Here are five related questions you might explore:
1. How do you differentiate the arctangent function with respect to variables other than \( y \)?
2. What are the general rules for taking partial derivatives of composite functions?
3. Can you apply this technique to other inverse trigonometric functions, such as \( \sin^{-1} \) or \( \cos^{-1} \)?
4. How would the derivative change if there were a different power of \( x \) in the function?
5. How do the results differ when taking the partial derivative with respect to \( x \) instead of \( y \)?

### Tip:
When differentiating composite functions, always remember to apply the chain rule, especially when working with inverse trigonometric functions.

Find the partial derivative of x^2 * tan^(-1)(y/x) with respect to y.

This problem involves finding the partial derivative of the function x^2 * tan^(-1)(y/x) with respect to y. The solution includes the use of the chain rule and the derivative of inverse trigonometric functions, and it is suitable for undergraduate calculus students.

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