Question

question 7 · 1 point what is r(k) when k = 3 and r(x)=(2x)^(6x)? select the correct answer below: o 6^18(6ln6 + 3) o 6^18(2ln6) o 18(6^17) o 6^18(6ln6 + 6)

Explanation:

Step1: Use power - rule and chain - rule

If $r(x)=(2x)^{6x}$, take the natural logarithm of both sides: $\ln r(x)=6x\ln(2x)$. Then differentiate both sides. The derivative of the left - hand side is $\frac{r^{\prime}(x)}{r(x)}$, and for the right - hand side, using the product rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = 6x$ and $v=\ln(2x)$. $u^\prime=6$ and $v^\prime=\frac{2}{2x}=\frac{1}{x}$. So $(\ln r(x))^\prime=\frac{r^{\prime}(x)}{r(x)}=6\ln(2x)+6x\cdot\frac{1}{x}=6\ln(2x) + 6$. Then $r^{\prime}(x)=r(x)(6\ln(2x)+6)=(2x)^{6x}(6\ln(2x)+6)$. When $x = k = 3$, $r^{\prime}(3)=(2\times3)^{6\times3}(6\ln(6)+6)=6^{18}(6\ln6 + 6)$.

Answer:

$6^{18}(6\ln6 + 6)$