Question

1. $f(x)=\sec ^{2}(x)$

s. $2 \sec ^{2} x \tan x$
c. $2 \sec x \tan x$
e. $\sec ^{2} x \tan x$
n. $\sec x$

Explanation:

Step1: Recall the chain rule

If \(y = u^{n}\), then \(y^\prime=nu^{n - 1}\cdot u^\prime\). Let \(u=\sec x\) and \(n = 2\).

Step2: Find the derivative of \(\sec x\)

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

Step3: Apply the chain rule

\(f(x)=\sec^{2}x\), using the chain rule \(f^\prime(x)=2\sec x\cdot(\sec x)^\prime\). Substitute \((\sec x)^\prime=\sec x\tan x\) into the formula. Then \(f^\prime(x)=2\sec x\cdot\sec x\tan x=2\sec^{2}x\tan x\)

Answer:

S. \(2\sec^{2}x\tan x\)