`sec^(-1)(x/a)`

Answer

To differentiate the function \( y = \sec^{-1}\left(\frac{x}{a}\right) \) with respect to \( x \), we will follow these steps:

### Step 1: Set up the equation
Let \( y = \sec^{-1}\left(\frac{x}{a}\right) \).

### Step 2: Differentiate using the chain rule
The derivative of \( \sec^{-1}(u) \) with respect to \( u \) is given by:
\[ \frac{d}{du} \sec^{-1}(u) = \frac{1}{|u| \sqrt{u^2 - 1}} \]
In our case, \( u = \frac{x}{a} \). Therefore, we need to find \( \frac{du}{dx} \).

### Step 3: Find \( \frac{du}{dx} \)
We differentiate \( u = \frac{x}{a} \):
\[ \frac{du}{dx} = \frac{1}{a} \]

### Step 4: Apply the chain rule
Now, using the chain rule:
\[ \frac{dy}{dx} = \frac{d}{du} \sec^{-1}(u) \cdot \frac{du}{dx} \]
Substituting \( u = \frac{x}{a} \):
\[ \frac{dy}{dx} = \frac{1}{\left|\frac{x}{a}\right| \sqrt{\left(\frac{x}{a}\right)^2 - 1}} \cdot \frac{1}{a} \]

### Step 5: Simplify the expression
Now we simplify:
\[ \frac{dy}{dx} = \frac{1}{\frac{|x|}{a} \sqrt{\frac{x^2}{a^2} - 1}} \cdot \frac{1}{a} \]
This can be rewritten as:
\[ \frac{dy}{dx} = \frac{a}{|x| \sqrt{\frac{x^2 - a^2}{a^2}}} \cdot \frac{1}{a} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1}{|x|} \cdot \sqrt{x^2 - a^2} \]

### Final Answer
Thus, the derivative of \( y = \sec^{-1}\left(\frac{x}{a}\right) \) is:
\[ \frac{dy}{dx} = \frac{1}{|x| \sqrt{x^2 - a^2}} \]