If ` y=(1)/(2) x sqrt""(x^2 +a^2 ) +1/2 a^2 log [x+ sqrt(x^2 +a^2 )]` , then `(dy)/(dx)`=

Options

A. `x sqrt""(x^2 +a^2)`
B. `1/2 sqrt""(x^2 +a^2)`
C. `sqrt""(x^2 +a^2)`
D. None of these

Answer

To find the derivative of the function

\[ y = \frac{1}{2} x \sqrt{x^2 + a^2} + \frac{1}{2} a^2 \log \left( x + \sqrt{x^2 + a^2} \right), \]

we will differentiate it step by step.

### Step 1: Differentiate the first term

The first term is

\[ \frac{1}{2} x \sqrt{x^2 + a^2}. \]

Using the product rule, where \( u = x \) and \( v = \sqrt{x^2 + a^2} \):

\[ \frac{dy_1}{dx} = \frac{1}{2} \left( u \frac{dv}{dx} + v \frac{du}{dx} \right). \]

Calculating \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):

- \( \frac{du}{dx} = 1 \)
- To find \( \frac{dv}{dx} \):

Using the chain rule,

\[ v = (x^2 + a^2)^{1/2} \implies \frac{dv}{dx} = \frac{1}{2}(x^2 + a^2)^{-1/2} \cdot (2x) = \frac{x}{\sqrt{x^2 + a^2}}. \]

Now substituting back into the product rule:

\[ \frac{dy_1}{dx} = \frac{1}{2} \left( x \cdot \frac{x}{\sqrt{x^2 + a^2}} + \sqrt{x^2 + a^2} \cdot 1 \right) = \frac{1}{2} \left( \frac{x^2}{\sqrt{x^2 + a^2}} + \sqrt{x^2 + a^2} \right). \]

### Step 2: Differentiate the second term

The second term is

\[ \frac{1}{2} a^2 \log \left( x + \sqrt{x^2 + a^2} \right). \]

Using the chain rule:

\[ \frac{dy_2}{dx} = \frac{1}{2} a^2 \cdot \frac{1}{x + \sqrt{x^2 + a^2}} \cdot \left( 1 + \frac{x}{\sqrt{x^2 + a^2}} \right). \]

The derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u = x + \sqrt{x^2 + a^2} \).

Calculating \( \frac{du}{dx} \):

\[ \frac{du}{dx} = 1 + \frac{x}{\sqrt{x^2 + a^2}}. \]

Thus,

\[ \frac{dy_2}{dx} = \frac{1}{2} a^2 \cdot \frac{1}{x + \sqrt{x^2 + a^2}} \cdot \left( 1 + \frac{x}{\sqrt{x^2 + a^2}} \right). \]

### Step 3: Combine the derivatives

Now, we combine both derivatives:

\[ \frac{dy}{dx} = \frac{1}{2} \left( \frac{x^2}{\sqrt{x^2 + a^2}} + \sqrt{x^2 + a^2} \right) + \frac{1}{2} a^2 \cdot \frac{1}{x + \sqrt{x^2 + a^2}} \cdot \left( 1 + \frac{x}{\sqrt{x^2 + a^2}} \right). \]

### Step 4: Simplify the expression

To simplify, we can factor out common terms and combine fractions as needed.

1. Factor out \( \frac{1}{2} \) from both terms.
2. Find a common denominator for the fractions.

After simplification, we will arrive at the final expression for \( \frac{dy}{dx} \).

### Final Result

The final result is:

\[ \frac{dy}{dx} = \frac{1}{2} \left( \frac{x^2 + a^2}{\sqrt{x^2 + a^2}} + \frac{a^2}{x + \sqrt{x^2 + a^2}} \cdot \left( 1 + \frac{x}{\sqrt{x^2 + a^2}} \right) \right). \]