Question

given the function $f(x)=2cos x$, determine the derivative of $f$ at $x = \frac{pi}{3}$ using the limit shown below. you do not have to simplify your answer. answer attempt 1 out of 2 $lim_{h
ightarrow0}$

Explanation:

Step1: Recall derivative limit - definition

The derivative of a function $f(x)$ at $x = a$ is given by $f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $f(x)=2\cos x$ and $a=\frac{\pi}{3}$.

Step2: Find $f(a + h)$ and $f(a)$

$f(\frac{\pi}{3}+h)=2\cos(\frac{\pi}{3}+h)$ and $f(\frac{\pi}{3}) = 2\cos\frac{\pi}{3}$.

Step3: Substitute into the limit - formula

$f^{\prime}(\frac{\pi}{3})=\lim_{h
ightarrow0}\frac{2\cos(\frac{\pi}{3}+h)-2\cos\frac{\pi}{3}}{h}$

Answer:

$\lim_{h
ightarrow0}\frac{2\cos(\frac{\pi}{3}+h)-2\cos\frac{\pi}{3}}{h}$