Differentiate `sqrt(x^(2)-sqrt((x^(2)-4)))`

Answer

To differentiate the function \( y = \sqrt{x^2 - \sqrt{x^2 - 4}} \), we will use the chain rule and the product rule of differentiation. Let's go through the steps systematically.

### Step 1: Rewrite the Function
We start with the function:
\[ y = \sqrt{x^2 - \sqrt{x^2 - 4}} \]
We can express this in terms of powers for easier differentiation:
\[ y = (x^2 - \sqrt{x^2 - 4})^{1/2} \]

### Step 2: Differentiate Using the Chain Rule
Using the chain rule, we differentiate \( y \):
\[ \frac{dy}{dx} = \frac{1}{2}(x^2 - \sqrt{x^2 - 4})^{-1/2} \cdot \frac{d}{dx}(x^2 - \sqrt{x^2 - 4}) \]

### Step 3: Differentiate the Inner Function
Next, we need to differentiate the inner function \( x^2 - \sqrt{x^2 - 4} \):
\[ \frac{d}{dx}(x^2 - \sqrt{x^2 - 4}) = 2x - \frac{1}{2}(x^2 - 4)^{-1/2} \cdot \frac{d}{dx}(x^2 - 4) \]
Now, differentiate \( x^2 - 4 \):
\[ \frac{d}{dx}(x^2 - 4) = 2x \]
Thus, substituting back, we have:
\[ \frac{d}{dx}(x^2 - \sqrt{x^2 - 4}) = 2x - \frac{1}{2}(x^2 - 4)^{-1/2} \cdot 2x \]
This simplifies to:
\[ \frac{d}{dx}(x^2 - \sqrt{x^2 - 4}) = 2x - \frac{x}{\sqrt{x^2 - 4}} \]

### Step 4: Substitute Back into the Derivative
Now we substitute this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{2}(x^2 - \sqrt{x^2 - 4})^{-1/2} \left( 2x - \frac{x}{\sqrt{x^2 - 4}} \right) \]

### Step 5: Simplify the Expression
We can simplify the expression further:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x^2 - \sqrt{x^2 - 4}}} \left( 2x - \frac{x}{\sqrt{x^2 - 4}} \right) \]

### Final Answer
Thus, the derivative of the function is:
\[ \frac{dy}{dx} = \frac{2x - \frac{x}{\sqrt{x^2 - 4}}}{2\sqrt{x^2 - \sqrt{x^2 - 4}}} \]