What is the derivative of `xsqrt(a^(2)-x^(2))+a^(2)sin^(-1)(x/a)` ?

Options

A. `sqrt(a^(2)-x^(2))`
B. `2sqrt(a^(2)-x^(2))`
C. `sqrt(x^(2)-a^(2))`
D. `2sqrt(x^(2)-a^(2))`

Answer

To find the derivative of the function \( f(x) = x \sqrt{a^2 - x^2} + a^2 \sin^{-1}\left(\frac{x}{a}\right) \), we will use the product rule and the chain rule of differentiation.

### Step-by-Step Solution:

1. **Identify the components of the function**:
- Let \( u = x \) and \( v = \sqrt{a^2 - x^2} \) for the first term \( x \sqrt{a^2 - x^2} \).
- The second term is \( a^2 \sin^{-1}\left(\frac{x}{a}\right) \).

2. **Differentiate the first term using the product rule**:
- The product rule states that \( \frac{d}{dx}(uv) = u'v + uv' \).
- Here, \( u' = \frac{d}{dx}(x) = 1 \) and \( v = \sqrt{a^2 - x^2} \).
- To find \( v' \):
\[ v' = \frac{d}{dx}(\sqrt{a^2 - x^2}) = \frac{1}{2\sqrt{a^2 - x^2}} \cdot (-2x) = \frac{-x}{\sqrt{a^2 - x^2}} \]
- Now apply the product rule:
\[ \frac{d}{dx}(x \sqrt{a^2 - x^2}) = 1 \cdot \sqrt{a^2 - x^2} + x \cdot \left(\frac{-x}{\sqrt{a^2 - x^2}}\right) \]
\[ = \sqrt{a^2 - x^2} - \frac{x^2}{\sqrt{a^2 - x^2}} \]
- Combine the terms:
\[ = \frac{(a^2 - x^2) - x^2}{\sqrt{a^2 - x^2}} = \frac{a^2 - 2x^2}{\sqrt{a^2 - x^2}} \]

3. **Differentiate the second term**:
- For the second term \( a^2 \sin^{-1}\left(\frac{x}{a}\right) \):
\[ \frac{d}{dx}\left(a^2 \sin^{-1}\left(\frac{x}{a}\right)\right) = a^2 \cdot \frac{1}{\sqrt{1 - \left(\frac{x}{a}\right)^2}} \cdot \frac{1}{a} = \frac{a^2}{\sqrt{1 - \frac{x^2}{a^2}}} \cdot \frac{1}{a} \]
\[ = \frac{a}{\sqrt{1 - \frac{x^2}{a^2}}} = \frac{a}{\sqrt{\frac{a^2 - x^2}{a^2}}} = \frac{a^2}{\sqrt{a^2 - x^2}} \]

4. **Combine the derivatives**:
- Now, combine the derivatives of both terms:
\[ f'(x) = \frac{a^2 - 2x^2}{\sqrt{a^2 - x^2}} + \frac{a^2}{\sqrt{a^2 - x^2}} = \frac{(a^2 - 2x^2) + a^2}{\sqrt{a^2 - x^2}} = \frac{2a^2 - 2x^2}{\sqrt{a^2 - x^2}} \]
- Factor out the common factor:
\[ = \frac{2(a^2 - x^2)}{\sqrt{a^2 - x^2}} \]

5. **Final result**:
- Thus, the derivative of the function is:
\[ f'(x) = 2\sqrt{a^2 - x^2} \]