Calculate the derivative of the function. Then find the value of the derivative as specified.
-$$\left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } \text { if } \mathrm { r } = \frac { 4 } { \sqrt { 12 - \theta } }$$
A) $\frac { \mathrm { dr } } { \mathrm { d } \theta } = - \frac { 2 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \frac { \mathrm { d } r } { \mathrm {~d} \theta } \mid \theta = 3 = - \frac { 2 } { 27 }$
B) $\frac { \mathrm { dr } } { \mathrm { d } \theta } = \frac { 2 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } = \frac { 2 } { 27 }$
C) $\frac { \mathrm { dr } } { \mathrm { d } \theta } = \frac { 4 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } = \frac { 4 } { 27 }$
D) $\frac { \mathrm { dr } } { \mathrm { d } \theta } = - \frac { 4 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \frac { \mathrm { dr } } { \mathrm { d } \theta } \mid \theta = 3 = - \frac { 4 } { 27 }$

Question

Quizplus · Accepted Answer

Using the power rule and the chain rule for derivatives, we get:

$$\frac { \mathrm { dr } } { \mathrm { d } \theta } = \frac { -4 } { 2\left(12-\theta\right)^{\frac {3}{2}}} = \frac { -2 } { \left(12-\theta\right)^{\frac {3}{2}}}$$

Substituting $\theta = 3$, we get:

$$\left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } = \frac {-2}{(12-3)^{\frac {3}{2}}} = \frac {-2}{27}$$

Therefore, the answer is B.

Calculate the Derivative of the Function drdθ∣θ=3 if r=412−θ\left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } \text { if } \mathrm { r } = \frac { 4 } { \sqrt { 12 - \theta } }dθdr​​θ=3​ if r=12−θ​4​

Calculate the Derivative of the Function $\left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } \text { if } \mathrm { r } = \frac { 4 } { \sqrt { 12 - \theta } }$