To find the derivative of the function \( f(x) = \arcsin\left(\frac{x}{y}\right) \) with respect to \(x\), we will apply the chain rule.

### Step-by-Step Solution:
1. **Understand the function**: We have \( f(x) = \arcsin\left(\frac{x}{y}\right) \), where \(y\) is a constant.

2. **Derivative of arcsin**: The derivative of \( \arcsin(u) \) with respect to \( u \) is given by:
   \[
   \frac{d}{du} \arcsin(u) = \frac{1}{\sqrt{1 - u^2}}
   \]
   where \( u \) is the argument of the arcsin function.

3. **Apply the chain rule**: Since \( u = \frac{x}{y} \), we apply the chain rule:
   \[
   \frac{d}{dx} \arcsin\left(\frac{x}{y}\right) = \frac{1}{\sqrt{1 - \left(\frac{x}{y}\right)^2}} \cdot \frac{d}{dx} \left(\frac{x}{y}\right)
   \]

4. **Differentiate \( \frac{x}{y} \) with respect to \(x\)**: Since \( y \) is a constant, the derivative of \( \frac{x}{y} \) with respect to \( x \) is:
   \[
   \frac{d}{dx} \left(\frac{x}{y}\right) = \frac{1}{y}
   \]

5. **Substitute and simplify**: Plugging this back into our chain rule expression, we get:
   \[
   \frac{d}{dx} \arcsin\left(\frac{x}{y}\right) = \frac{1}{\sqrt{1 - \left(\

Learn how to find the derivative of arcsin(x/y) step-by-step. This problem covers differentiation of inverse trigonometric functions and application of the chain rule in calculus. Suitable for advanced high school students and calculus enthusiasts.

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