Differentiate the following functions w.r.t. x:
`(sec ( ax-b) )/( x^(2) -2)`

Answer

To differentiate the function \( f(x) = \frac{\sec(ax - b)}{x^2 - 2} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

### Step 1: Identify \( u \) and \( v \)
Here, we identify:
- \( u = \sec(ax - b) \)
- \( v = x^2 - 2 \)

### Step 2: Differentiate \( u \) and \( v \)
Next, we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

**Differentiating \( u \):**
Using the chain rule, we differentiate \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(\sec(ax - b)) = \sec(ax - b) \tan(ax - b) \cdot \frac{d}{dx}(ax - b) = \sec(ax - b) \tan(ax - b) \cdot a \]
Thus,
\[ \frac{du}{dx} = a \sec(ax - b) \tan(ax - b) \]

**Differentiating \( v \):**
Now, we differentiate \( v \):
\[ \frac{dv}{dx} = \frac{d}{dx}(x^2 - 2) = 2x \]

### Step 3: Substitute into the Quotient Rule
Now we substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{(x^2 - 2)(a \sec(ax - b) \tan(ax - b)) - (\sec(ax - b))(2x)}{(x^2 - 2)^2} \]

### Step 4: Simplify the Expression
We can simplify the expression:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{a(x^2 - 2) \sec(ax - b) \tan(ax - b) - 2x \sec(ax - b)}{(x^2 - 2)^2} \]
Factoring out \( \sec(ax - b) \):
\[ = \frac{\sec(ax - b) \left[ a(x^2 - 2) \tan(ax - b) - 2x \right]}{(x^2 - 2)^2} \]

### Final Answer
Thus, the derivative of the function \( f(x) = \frac{\sec(ax - b)}{x^2 - 2} \) is:
\[ f'(x) = \frac{\sec(ax - b) \left[ a(x^2 - 2) \tan(ax - b) - 2x \right]}{(x^2 - 2)^2} \]