Certainly! To find the derivative of [tex]\(\sec^2(x)\)[/tex], follow these steps:
1. Identify the function to be differentiated: The function we're dealing with is [tex]\(\sec^2(x)\)[/tex].
2. Rewrite the function using the chain rule: The chain rule is often used when differentiating composite functions. Here, we can write [tex]\(\sec^2(x)\)[/tex] as [tex]\((\sec(x))^2\)[/tex].
3. Differentiate the outer function: Let [tex]\(u = \sec(x)\)[/tex]. Then our function becomes [tex]\(u^2\)[/tex]. The derivative of [tex]\(u^2\)[/tex] with respect to [tex]\(u\)[/tex] is [tex]\(2u\)[/tex].
4. Differentiate the inner function: Next, we need to differentiate [tex]\(u = \sec(x)\)[/tex] with respect to [tex]\(x\)[/tex]. The derivative of [tex]\(\sec(x)\)[/tex] is [tex]\(\sec(x) \tan(x)\)[/tex].
5. Apply the chain rule: Multiply the derivative of the outer function by the derivative of the inner function:
[tex]\[
\frac{d}{dx} (\sec^2(x)) = 2 \sec(x) \cdot \sec(x) \tan(x)
\][/tex]
6. Simplify the resulting expression: Combining these together, we get:
[tex]\[
\frac{d}{dx} (\sec^2(x)) = 2 \sec(x) \cdot \sec(x) \tan(x) = 2 \sec^2(x) \tan(x)
\][/tex]
Therefore, the derivative of [tex]\(\sec^2(x)\)[/tex] is:
[tex]\[
\boxed{2 \sec^2(x) \tan(x)}
\][/tex]
