Solve the problem.
-Evaluate $\frac { \mathrm { dw } } { \mathrm { dt } }$ at $\mathrm { t } = \frac { 5 } { 2 } \pi$ for the function $\mathrm { w } = \frac { \mathrm { xy } } { \mathrm { z } } ; \mathrm { x } = \sin \mathrm { t } , \mathrm { y } = \cos \mathrm { t } , \mathrm { z } = \mathrm { t } ^ { 2 }$.
A) $- \frac { 4 } { 25 } \left( \frac { 1 } { \pi ^ { 2 } } \right)$
B) $\frac { 1 } { 2 } \left( \frac { 1 } { \pi ^ { 2 } } \right)$
C) $- \frac { 1 } { 2 } \left( \frac { 1 } { \pi ^ { 2 } } \right)$
D) $- \frac { 1 } { 2 } \left( \frac { 1 } { \pi } \right)$

Question

Quizplus · Accepted Answer

To find $\frac { \mathrm { dw } } { \mathrm { dt } }$ for $w = \frac { xy } { z }$ with $x = \sin t$, $y = \cos t$, and $z = t^2$, we first substitute $x$, $y$, and $z$ into $w$ to get $w = \frac{\sin t \cdot \cos t}{t^2}$. Then, we differentiate $w$ with respect to $t$ using the quotient and product rules. After differentiation, we substitute $t = \frac{5}{2}\pi$ into the derivative to find the specific value. The correct answer involves calculating this derivative correctly and evaluating it at the given $t$ value, leading to the answer choice A.

Solve the Problem dwdt\frac { \mathrm { dw } } { \mathrm { dt } }dtdw​

Solve the Problem $\frac { \mathrm { dw } } { \mathrm { dt } }$