To find the derivative of $\sec^2(x)$, follow these steps:

1. Recall that $\sec^2(x)$ is a composite function: $(\sec(x))^2$. We will use the chain rule and the derivative of $\sec(x)$.
2. The chain rule states that the derivative of $(f(x))^n$ is $n \cdot (f(x))^{n-1} \cdot f'(x)$.

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

3. The derivative of $\sec(x)$ is $\sec(x)\tan(x)$.

Thus:

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

4. Simplify the expression:

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

Final Answer:

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

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Related Questions:

1. What is the derivative of $\sec(x)$?
2. How does the chain rule work in other trigonometric derivatives?
3. What is the integral of $\sec^2(x)$?
4. Can you apply this derivative to find the slope of a tangent line to $\sec^2(x)$?
5. How does the derivative of $\tan(x)$ relate to this function?

Tip:

Always keep a list of common derivatives and integrals of trigonometric functions handy—they appear frequently in calculus problems!