Question

attempt 2: 1 attempt remaining. determine $g(5)$ if $g(s)=arctan(s)$. $g(5)=$

Explanation:

Step1: Recall the second - derivative formula for $y = \arctan(s)$

The first - derivative of $y=\arctan(s)$ is $y'=\frac{1}{1 + s^{2}}$ by the formula $\frac{d}{ds}\arctan(s)=\frac{1}{1 + s^{2}}$.

Step2: Use the quotient rule to find the second - derivative

The quotient rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 1$, $u'=0$, $v=1 + s^{2}$, and $v' = 2s$. So, $y''=\frac{0\times(1 + s^{2})-1\times2s}{(1 + s^{2})^{2}}=-\frac{2s}{(1 + s^{2})^{2}}$.

Step3: Evaluate the second - derivative at $s = 5$

Substitute $s = 5$ into $y''$. We get $y''(5)=-\frac{2\times5}{(1 + 5^{2})^{2}}=-\frac{10}{(1 + 25)^{2}}=-\frac{10}{26^{2}}=-\frac{10}{676}=-\frac{5}{338}$.

Answer:

$-\frac{5}{338}$