If : `f(x)=(1)/(sqrt(x^(2)+a^(2))+sqrt(x^(2)+b^(2)))" then: "f'(x)=`

Options

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

Answer

To find the derivative \( f'(x) \) of the function

\[ f(x) = \frac{1}{\sqrt{x^2 + a^2} + \sqrt{x^2 + b^2}}, \]

we will follow these steps:

### Step 1: Simplify the function
We will rationalize the denominator. To do this, we multiply the numerator and denominator by the conjugate of the denominator:

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

### Step 2: Multiply out the numerator and denominator
The numerator becomes:

\[ \sqrt{x^2 + a^2} - \sqrt{x^2 + b^2}. \]

The denominator simplifies as follows:

\[ (\sqrt{x^2 + a^2})^2 - (\sqrt{x^2 + b^2})^2 = (x^2 + a^2) - (x^2 + b^2) = a^2 - b^2. \]

Thus, we have:

\[ f(x) = \frac{\sqrt{x^2 + a^2} - \sqrt{x^2 + b^2}}{a^2 - b^2}. \]

### Step 3: Differentiate \( f(x) \)
Now we differentiate \( f(x) \):

\[ f'(x) = \frac{1}{a^2 - b^2} \left( \frac{d}{dx}(\sqrt{x^2 + a^2}) - \frac{d}{dx}(\sqrt{x^2 + b^2}) \right). \]

Using the chain rule, we find the derivatives:

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

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

### Step 4: Substitute the derivatives back into \( f'(x) \)
Now substituting back, we get:

\[ f'(x) = \frac{1}{a^2 - b^2} \left( \frac{x}{\sqrt{x^2 + a^2}} - \frac{x}{\sqrt{x^2 + b^2}} \right). \]

### Step 5: Factor out \( x \)
Factoring out \( x \) gives us:

\[ f'(x) = \frac{x}{a^2 - b^2} \left( \frac{1}{\sqrt{x^2 + a^2}} - \frac{1}{\sqrt{x^2 + b^2}} \right). \]

### Final Result
Thus, the derivative \( f'(x) \) is:

\[ f'(x) = \frac{x}{a^2 - b^2} \left( \frac{1}{\sqrt{x^2 + a^2}} - \frac{1}{\sqrt{x^2 + b^2}} \right). \]

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