Q. \(\frac{d}{dx}\sec^2(x)\)

We want to find the derivative of \( \sec^{2}(x) \). This means we are differentiating the function where the base is \( \sec(x) \) and the whole quantity is squared.

Step 1: Identify the outer and inner functions

Let \(u = \sec(x)\). Then the given function becomes

\[
\sec^{2}(x) = ( \sec(x) )^{2} = u^{2}.
\]

So we are differentiating \(u^{2}\) with respect to \(x\).

Step 2: Differentiate the outer function

Using the power rule, for \(u^{2}\) we have

\[
\frac{d}{dx}\left(u^{2}\right) = 2u \cdot \frac{du}{dx}.
\]

Step 3: Differentiate the inner function

Now we must compute \( \frac{du}{dx} \) where \(u = \sec(x)\).

Recall the derivative rule:

\[
\frac{d}{dx}\left(\sec(x)\right) = \sec(x)\tan(x).
\]

Therefore,

\[
\frac{du}{dx} = \sec(x)\tan(x).
\]

Step 4: Combine using the chain rule

Substitute \(u = \sec(x)\) and \( \frac{du}{dx} = \sec(x)\tan(x) \) into

\[
\frac{d}{dx}\left(u^{2}\right) = 2u \cdot \frac{du}{dx}.
\]

We get:

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

Step 5: Simplify

Multiply the factors \( \sec(x)\cdot \sec(x) \) to get \( \sec^{2}(x) \):

\[
2\sec(x)\left(\sec(x)\tan(x)\right) = 2\sec^{2}(x)\tan(x).
\]

Final Answer

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