homework5: problem 41 (1 point) compute the first four derivatives of $g(x)=2cos x$. a. $g(x)=$ b. $g(x)=$…

homework5: problem 41 (1 point) compute the first four derivatives of $g(x)=2cos x$. a. $g(x)=$ b. $g(x)=$ c. $g(x)=$ d. $g^{(4)}(x)=$ note: you can earn partial credit on this problem. preview my answers submit answers you have attempted this problem 0 times. you have unlimited attempts remaining.

homework5: problem 41 (1 point) compute the first four derivatives of $g(x)=2cos x$. a. $g(x)=$ b. $g(x)=$ c. $g(x)=$ d. $g^{(4)}(x)=$ note: you can earn partial credit on this problem. preview my answers submit answers you have attempted this problem 0 times. you have unlimited attempts remaining.

Answer

Explanation:

Step1: Recall derivative of cosine

The derivative of $\cos x$ is $-\sin x$. Using the constant - multiple rule $(cf(x))' = cf'(x)$ where $c = 2$ and $f(x)=\cos x$, we have $g'(x)=2(-\sin x)=- 2\sin x$.

Step2: Find second - derivative

The derivative of $\sin x$ is $\cos x$. So, $g''(x)=-2(\cos x)=-2\cos x$.

Step3: Find third - derivative

Since the derivative of $\cos x$ is $-\sin x$, $g'''(x)=-2(-\sin x)=2\sin x$.

Step4: Find fourth - derivative

The derivative of $\sin x$ is $\cos x$. So, $g^{(4)}(x)=2\cos x$.

Answer:

a. $-2\sin x$ b. $-2\cos x$ c. $2\sin x$ d. $2\cos x$