deterimine y' if y= x/ square root of 1-x^2

Given the function \( y = \frac{x}{\sqrt{1 - x^2}} \), we want to find its derivative \( y' \).

### Step-by-Step Solution:

1. **Rewrite the function:**

   \[
   y = x \cdot (1 - x^2)^{-\frac{1}{2}}
   \]

   This allows us to use the product rule for differentiation.

2. **Differentiate using the product rule:**

   The product rule states that if \( y = u \cdot v \), then \( y' = u' \cdot v + u \cdot v' \).

   Here, \( u = x \) and \( v = (1 - x^2)^{-\frac{1}{2}} \).

   - First, find \( u' \):
     \[
     u' = \frac{d}{dx}(x) = 1
     \]

   - Next, find \( v' \):
     \[
     v' = \frac{d}{dx}\left((1 - x^2)^{-\frac{1}{2}}\right)
     \]
     To differentiate \( v \), use the chain rule. Let \( g(x) = 1 - x^2 \) and \( h(g) = g^{-\frac{1}{2}} \). Then,
     \[
     v' = h'(g(x)) \cdot g'(x) = -\frac{1}{2}(1 - x^2)^{-\frac{3}{2}} \cdot (-2x)
     \]
     Simplifying,
     \[
     v' = \frac{x}{(1 - x^2)^{\frac{3}{2}}}
     \]

3. **Apply the product rule:**

   \[
   y' = u' \cdot v + u \cdot v'
   \]
   Substituting the values of \( u' \), \( v \), and \( v' \):
   \[
   y' = 1 \cdot (1 - x^2)^{-\frac{1}{2}} + x \cdot \frac{x}{(1 - x^2)^{\frac{3}{2}}}
   \]

   Simplifying further:
   \[
   y' = \frac{1}{\sqrt{1 - x^2}} + \frac{x^2}{(1 - x^2)^{\frac{3}{2}}}
   \]

   Combine the terms over a common denominator:
   \[
   y' = \frac{(1 - x^2) + x^2}{(1 - x^2)^{\frac{3}{2}}}
   \]
   \[
   y' = \frac{1}{(1 - x^2)^{\frac{3}{2}}}
   \]

### Final Answer:
\[
y' = \frac{1}{(1 - x^2)^{\frac{3}{2}}}
\]

Would you like more details or have any questions about this process?

### Related Questions:
1. How do you apply the product rule in differentiation?
2. What is the chain rule in differentiation, and how is it used?
3. How do you simplify expressions with exponents after differentiation?
4. Can you provide examples of differentiating other functions involving square roots?
5. How do you differentiate functions with products and quotients simultaneously?
6. What are some common mistakes to avoid when using the product rule?
7. How does the derivative of a function change as its graph changes?
8. How do you interpret the derivative of a function in real-world applications?

### Tip:
Always simplify the function as much as possible before applying differentiation rules to make the process easier.

Learn how to find the derivative of the function y = x/sqrt(1 - x^2) using differentiation rules. This problem involves applying the product rule and the chain rule, suitable for advanced high school and college students.

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