Question

evaluate the limit and write your answer in simplest form: $lim_{h
ightarrow0}\frac{2csc(\frac{7pi}{4}+h)-2csc(\frac{7pi}{4})}{h}$

Explanation:

Step1: Recall the definition of the derivative

The given limit $\lim_{h
ightarrow0}\frac{2\csc(\frac{7\pi}{4}+h)-2\csc(\frac{7\pi}{4})}{h}$ is in the form of the derivative definition $f^\prime(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}$, where $f(x)=2\csc(x)$ and $x = \frac{7\pi}{4}$.

Step2: Find the derivative of $y = 2\csc(x)$

We know that the derivative of $\csc(x)$ is $-\csc(x)\cot(x)$. Using the constant - multiple rule, if $y = 2\csc(x)$, then $y^\prime=- 2\csc(x)\cot(x)$.

Step3: Evaluate the derivative at $x=\frac{7\pi}{4}$

First, find $\csc(\frac{7\pi}{4})$ and $\cot(\frac{7\pi}{4})$.
We know that $\sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2}$, so $\csc(\frac{7\pi}{4})=-\sqrt{2}$.
Also, $\tan(\frac{7\pi}{4})=- 1$, so $\cot(\frac{7\pi}{4})=-1$.
Then $y^\prime\big|_{x = \frac{7\pi}{4}}=-2\times(-\sqrt{2})\times(-1)=-2\sqrt{2}$.

Answer:

$-2\sqrt{2}$