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

Options

A. `sqrt(x^(2) +a^(2))`
B. `1/sqrt(x^(2) +a^(2))`
C. `2sqrt(x^(2) +a^(2))`
D. `2/sqrt(x^(2) +a^(2))`

Answer

To find the derivative \( \frac{dy}{dx} \) for the given function

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

we will differentiate each term separately.

### Step 1: Differentiate the first term

The first term is

\[ y_1 = \frac{x}{2 \sqrt{a^2 + x^2}}. \]

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

\[ \frac{dy_1}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}. \]

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

- \( \frac{du}{dx} = 1 \)
- \( v = 2\sqrt{a^2 + x^2} \)
- \( \frac{dv}{dx} = 2 \cdot \frac{1}{2\sqrt{a^2 + x^2}} \cdot 2x = \frac{2x}{\sqrt{a^2 + x^2}} \)

Now substituting back into the quotient rule:

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

This simplifies to:

\[ \frac{dy_1}{dx} = \frac{2\sqrt{a^2 + x^2} - \frac{2x^2}{\sqrt{a^2 + x^2}}}{4(a^2 + x^2)} = \frac{2(a^2 + x^2) - 2x^2}{4\sqrt{a^2 + x^2}(a^2 + x^2)} = \frac{2a^2}{4\sqrt{a^2 + x^2}(a^2 + x^2)} = \frac{a^2}{2\sqrt{a^2 + x^2}(a^2 + x^2)}. \]

### Step 2: Differentiate the second term

The second term is

\[ y_2 = \frac{a^2}{2} \log\left(x + \sqrt{x^2 + a^2}\right). \]

Using the chain rule:

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

The derivative of \( \sqrt{x^2 + a^2} \) is \( \frac{x}{\sqrt{x^2 + a^2}} \).

Thus,

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

Combining the two derivatives:

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

### Final Answer

Thus, the derivative \( \frac{dy}{dx} \) is:

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