Let's simplify the derivative further.

We previously found the derivative:
\[
f'(x) = 2x\sqrt{1 - x^2} - \frac{x^3}{\sqrt{1 - x^2}}
\]

### Simplification:

1. **Factor out the common term:**
   
   Both terms in the expression share a common factor of \( x \). Factor \( x \) out:
   \[
   f'(x) = x \left( 2\sqrt{1 - x^2} - \frac{x^2}{\sqrt{1 - x^2}} \right)
   \]

2. **Combine the terms inside the parentheses:**

   To combine the terms inside the parentheses, find a common denominator. The common denominator here would be \( \sqrt{1 - x^2} \):
   \[
   f'(x) = x \left( \frac{2(1 - x^2) - x^2}{\sqrt{1 - x^2}} \right)
   \]
   Simplify the numerator:
   \[
   2(1 - x^2) - x^2 = 2 - 2x^2 - x^2 = 2 - 3x^2
   \]

   So the expression becomes:
   \[
   f'(x) = x \cdot \frac{2 - 3x^2}{\sqrt{1 - x^2}}
   \]

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

This is the simplified form of the derivative.

Would you like more details or have any questions?

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### Related Questions:
1. What are the critical points of the function \( f(x) = x^2 \cdot \sqrt{1 - x^2} \)?
2. How do you interpret the physical meaning of this derivative?
3. What is the second derivative of the function?
4. How would you find the maximum and minimum values of \( f(x) \)?
5. Can you explain how to simplify more complex derivatives?

### Tip:
Always look for common factors to simplify expressions, especially when dealing with products and sums involving square roots or fractions.

Learn how to simplify the derivative 2x sqrt(1 - x^2) - x^3 / sqrt(1 - x^2) step-by-step. This problem involves differentiation and algebraic manipulation, focusing on factoring out common terms and simplifying complex expressions. Suitable for advanced high school and college-level students.

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