overall accuracy: 85.7% record: 20 score: 20 evaluate lim(x→π/6) (cos(x) - cos(π/6))/(x - π/6) -√3/2 -1/2…

Explanation:

Step1: Recall derivative definition

The limit $\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x - a}$ is the definition of the derivative of $y = f(x)$ at $x=a$. Here $f(x)=\cos(x)$ and $a = \frac{\pi}{6}$.

Step2: Find derivative of cosine

The derivative of $y=\cos(x)$ is $y'=-\sin(x)$.

Step3: Evaluate derivative at $x = \frac{\pi}{6}$

Substitute $x=\frac{\pi}{6}$ into $y'=-\sin(x)$. We get $-\sin(\frac{\pi}{6})=-\frac{1}{2}$.

Answer:

$-\frac{1}{2}$