if (f(x)=\tan x), then (lim_{x\rightarrow\frac{pi}{4}}\frac{f(x)-f(\frac{pi}{4})}{x - \frac{pi}{4}}) is\na…

Explanation:

Step1: Recall the definition of the derivative

The given limit $\lim_{x\rightarrow\frac{\pi}{4}}\frac{f(x)-f(\frac{\pi}{4})}{x - \frac{\pi}{4}}$ is in the form of the derivative of the function $y = f(x)$ at $x=a$, where $a=\frac{\pi}{4}$ and $f(x)=\tan x$. The derivative of a function $y = f(x)$ is defined as $f^\prime(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x - a}$.

Step2: Find the derivative of $f(x)=\tan x$

The derivative of $y = \tan x$ with respect to $x$ is $y^\prime=\sec^{2}x$ (using the derivative formula $\frac{d}{dx}(\tan x)=\sec^{2}x$).

Step3: Evaluate the derivative at $x = \frac{\pi}{4}$

We need to find the value of $f^\prime(x)$ at $x=\frac{\pi}{4}$. Substitute $x = \frac{\pi}{4}$ into $y^\prime=\sec^{2}x$. So $f^\prime(\frac{\pi}{4})=\sec^{2}(\frac{\pi}{4})$.

Answer:

C. $\sec^{2}(\frac{\pi}{4})$