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The given problem is asking to differentiate the function \( y = \frac{1}{\sqrt{a + x^2}} \), with the substitution \( u = a + x^2 \).

Let me guide you through the steps:

### Step-by-Step Solution:
1. First, rewrite the equation using the substitution \( u = a + x^2 \):
   \[
   y = \frac{1}{\sqrt{u}}.
   \]
   
2. Now, apply the chain rule. The derivative of \( y \) with respect to \( u \) is:
   \[
   \frac{dy}{du} = -\frac{1}{2}u^{-\frac{3}{2}}.
   \]
   
3. Next, differentiate \( u \) with respect to \( x \):
   \[
   \frac{du}{dx} = 2x.
   \]

4. Finally, apply the chain rule:
   \[
   \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\frac{1}{2}u^{-\frac{3}{2}} \cdot 2x = -\frac{x}{(a + x^2)^{3/2}}.
   \]

Thus, the derivative of \( y = \frac{1}{\sqrt{a + x^2}} \) is:
\[
\frac{dy}{dx} = -\frac{x}{(a + x^2)^{3/2}}.
\]

Would you like more details on any part of this solution?

### Related Questions:
1. How do you apply the chain rule in different types of functions?
2. What are the implications of differentiating a square root function?
3. How do you handle higher-order derivatives for this function?
4. Can you apply implicit differentiation to functions involving square roots?
5. How do you integrate a function like \( \frac{1}{(a + x^2)^{3/2}} \)?

**Tip:** When differentiating composite functions, always consider the chain rule as a powerful method to simplify the process.

Differentiate y = 1 / √(a + x²), with the substitution u = a + x².

This guide explains how to differentiate the function y = 1 / √(a + x²) using substitution and the chain rule. Follow step-by-step instructions to apply differentiation to composite functions. Suitable for advanced calculus students in Grades 11-12.

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