Differentiate 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 apply the rules of differentiation step by step.

### Step 1: Differentiate \( e^{\sec^2 x} \)

Using the chain rule, we differentiate \( e^{u} \) where \( u = \sec^2 x \).

\[ \frac{dy}{dx} = e^{\sec^2 x} \cdot \frac{d}{dx}(\sec^2 x) \]

Now, we need to differentiate \( \sec^2 x \).

### Step 2: Differentiate \( \sec^2 x \)

Using the chain rule again, we know that:

\[ \frac{d}{dx}(\sec^2 x) = 2\sec^2 x \tan x \]

So, substituting this back, we have:

\[ \frac{dy}{dx} = e^{\sec^2 x} \cdot (2\sec^2 x \tan x) \]

### Step 3: Differentiate \( 3\cos^{-1} x \)

Next, we differentiate \( 3\cos^{-1} x \). The derivative of \( \cos^{-1} x \) is:

\[ \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}} \]

Thus, the derivative of \( 3\cos^{-1} x \) is:

\[ \frac{d}{dx}(3\cos^{-1} x) = 3 \cdot \left(-\frac{1}{\sqrt{1 - x^2}}\right) = -\frac{3}{\sqrt{1 - x^2}} \]

### Step 4: Combine the derivatives

Now, we can combine the derivatives from Steps 2 and 3:

\[ \frac{dy}{dx} = e^{\sec^2 x} \cdot (2\sec^2 x \tan x) - \frac{3}{\sqrt{1 - x^2}} \]

### Final Answer

Thus, the derivative of the function \( y = e^{\sec^2 x} + 3\cos^{-1} x \) with respect to \( x \) is:

\[ \frac{dy}{dx} = 2\sec^2 x \tan x \cdot e^{\sec^2 x} - \frac{3}{\sqrt{1 - x^2}} \]

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