Differentiate the following functions w.r.t. x:
`sec ax`

Answer

To differentiate the function \( y = \sec(ax) \) with respect to \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the function to differentiate**:
\[ y = \sec(ax) \]

2. **Recall the differentiation formula for secant**:
The derivative of \( \sec(u) \) with respect to \( u \) is given by:
\[ \frac{d}{du}(\sec(u)) = \sec(u) \tan(u) \]
where \( u = ax \).

3. **Apply the chain rule**:
Since \( u = ax \), we need to apply the chain rule. The derivative of \( y \) with respect to \( x \) will be:
\[ \frac{dy}{dx} = \frac{d}{du}(\sec(u)) \cdot \frac{du}{dx} \]

4. **Differentiate \( u = ax \)**:
The derivative of \( u = ax \) with respect to \( x \) is:
\[ \frac{du}{dx} = a \]

5. **Combine the derivatives**:
Now substituting back, we have:
\[ \frac{dy}{dx} = \sec(ax) \tan(ax) \cdot a \]

6. **Final result**:
Therefore, the derivative of \( y = \sec(ax) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = a \sec(ax) \tan(ax) \]

### Conclusion:
The final answer is:
\[ \frac{dy}{dx} = a \sec(ax) \tan(ax) \]