To differentiate the function [tex]\( f(x) = \tan^2(x) \)[/tex], follow these steps:
1. Rewrite the function in a more convenient form:
[tex]\[
f(x) = (\tan(x))^2
\][/tex]
2. Apply the chain rule:
The chain rule states that if you have a composite function [tex]\( (g(h(x))) \)[/tex], the derivative is:
[tex]\[
\frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x)
\][/tex]
For our function, [tex]\( g(u) = u^2 \)[/tex] and [tex]\( u = \tan(x) \)[/tex]. So we first need to find the derivatives of these components.
3. Differentiate the outer function [tex]\( g(u) = u^2 \)[/tex]:
The derivative of [tex]\( u^2 \)[/tex] with respect to [tex]\( u \)[/tex] is:
[tex]\[
g'(u) = 2u
\][/tex]
4. Differentiate the inner function [tex]\( u = \tan(x) \)[/tex]:
The derivative of [tex]\( \tan(x) \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{d}{dx}[\tan(x)] = \sec^2(x)
\][/tex]
5. Combine the results using the chain rule:
[tex]\[
\frac{d}{dx}[\tan^2(x)] = g'(u) \cdot \frac{d}{dx}[u]
\][/tex]
Substitute [tex]\( g'(u) \)[/tex] and [tex]\( \frac{d}{dx}[u] \)[/tex]:
[tex]\[
\frac{d}{dx}[\tan^2(x)] = 2\tan(x) \cdot \sec^2(x)
\][/tex]
6. Simplify the expression:
Recall that [tex]\( \sec(x) = \frac{1}{\cos(x)} \)[/tex], so:
[tex]\[
\sec^2(x) = \frac{1}{\cos^2(x)}
\][/tex]
Thus,
[tex]\[
\frac{d}{dx}[\tan^2(x)] = 2\tan(x) \cdot \sec^2(x)
\][/tex]
Since [tex]\( \sec^2(x) = 1 + \tan^2(x) \)[/tex], we can substitute this to get a more detailed expression:
[tex]\[
\frac{d}{dx}[\tan^2(x)] = 2\tan(x) \cdot (1 + \tan^2(x))
\][/tex]
7. Final expression:
The derivative of [tex]\( \tan^2(x) \)[/tex] is:
[tex]\[
\frac{d}{dx}[\tan^2(x)] = (2\tan(x))(1 + \tan^2(x))
\][/tex]
Which can also be written as:
[tex]\[
\frac{d}{dx}[\tan^2(x)] = (2\tan(x))(1 + \tan^2(x)) = (2\tan(x)\tan^2(x) + 2\tan(x))
\][/tex]
Thus, the final derivative is:
[tex]\[
\frac{d}{dx}[\tan^2(x)] = (2\tan(x)^2 + 2)\tan(x)
\][/tex]
