Differentiable the following w.r.t. x :
`e^(sec^(2)x)+3cos^(-1)x`.

Answer

To differentiate the function \( y = e^{\sec^2 x} + 3 \cos^{-1} x \) with respect to \( x \), we will use the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.

### Step-by-Step Solution:

1. **Identify the components of the function**:
\[ y = e^{\sec^2 x} + 3 \cos^{-1} x \]
Here, we can denote:
- \( u = e^{\sec^2 x} \)
- \( v = 3 \cos^{-1} x \)

2. **Differentiate \( u \)**:
To differentiate \( u = e^{\sec^2 x} \), we will use the chain rule. The derivative of \( e^f \) is \( e^f \cdot f' \).

- First, find \( f = \sec^2 x \).
- The derivative \( f' = \frac{d}{dx}(\sec^2 x) \) can be found using the chain rule:
\[ \frac{d}{dx}(\sec^2 x) = 2 \sec^2 x \tan x \]
- Therefore, using the chain rule:
\[ \frac{du}{dx} = e^{\sec^2 x} \cdot \frac{d}{dx}(\sec^2 x) = e^{\sec^2 x} \cdot 2 \sec^2 x \tan x \]

3. **Differentiate \( v \)**:
The derivative of \( v = 3 \cos^{-1} x \) is:
\[ \frac{dv}{dx} = 3 \cdot \frac{d}{dx}(\cos^{-1} x) = 3 \cdot \left(-\frac{1}{\sqrt{1 - x^2}}\right) = -\frac{3}{\sqrt{1 - x^2}} \]

4. **Combine the derivatives**:
Now, we can combine the derivatives using the sum rule:
\[ \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = e^{\sec^2 x} \cdot 2 \sec^2 x \tan x - \frac{3}{\sqrt{1 - x^2}} \]

5. **Final expression**:
Thus, the derivative of the given function with respect to \( x \) is:
\[ \frac{dy}{dx} = 2 \sec^2 x \tan x \cdot e^{\sec^2 x} - \frac{3}{\sqrt{1 - x^2}} \]