Find the second derivative of the function.
-$$r = \left( \frac { 1 + 8 \theta } { 8 \theta } \right) ( 8 - \theta )$$
A) $\frac { \mathrm { d } ^ { 2 } r } { d \theta ^ { 2 } } = \frac { 2 } { \theta ^ { 3 } }$
B) $\frac { \mathrm { d } ^ { 2 } r } { d \theta ^ { 2 } } = \frac { 1 } { \theta } - \theta$
C) $\frac { d ^ { 2 } r } { d \theta ^ { 2 } } = - \frac { 2 } { \theta ^ { 3 } } - 1$
D) $\frac { d ^ { 2 } r } { d \theta ^ { 2 } } = - \frac { 1 } { \theta ^ { 2 } } - 1$

Question

Quizplus · Accepted Answer

First, simplify the given function $r = \left( \frac { 1 + 8 \theta } { 8 \theta } \right) ( 8 - \theta )$ to $r = \left( \frac { 1 } { 8 \theta } + 1 \right) ( 8 - \theta )$, which simplifies further to $r = 1 + \frac{1}{\theta} - \frac{\theta}{8} - \frac{1}{8}$. The first derivative $r' = -\frac{1}{\theta^2} - \frac{1}{8}$, and the second derivative $r'' = \frac{2}{\theta^3}$.

Find the Second Derivative of the Function r=(1+8θ8θ)(8−θ)r = \left( \frac { 1 + 8 \theta } { 8 \theta } \right) ( 8 - \theta )r=(8θ1+8θ​)(8−θ)

Find the Second Derivative of the Function $r = \left( \frac { 1 + 8 \theta } { 8 \theta } \right) ( 8 - \theta )$