Question

attempt 2: 1 attempt remaining. (1 point) if $f(x)=2arctan(8x)$, find $f(x)$. find $f(3)$. submit answer next item

Explanation:

Step1: Recall derivative formula

The derivative of $\arctan(u)$ with respect to $x$ is $\frac{u'}{1 + u^{2}}$. Here $u = 8x$ and the function is $y=2\arctan(8x)$. By the constant - multiple rule of differentiation $(cf(x))'=cf'(x)$ where $c = 2$.

Step2: Differentiate $u = 8x$

The derivative of $u = 8x$ with respect to $x$ is $u'=8$.

Step3: Apply the formula for $y'$

Using the formula for the derivative of $\arctan(u)$ and the constant - multiple rule, we have $f'(x)=2\times\frac{8}{1+(8x)^{2}}=\frac{16}{1 + 64x^{2}}$.

Step4: Find $f'(3)$

Substitute $x = 3$ into $f'(x)$. So $f'(3)=\frac{16}{1+64\times3^{2}}=\frac{16}{1 + 576}=\frac{16}{577}$.

Answer:

$\frac{16}{577}$