The resistance R of a certain type of resistor is $R = \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } ,$ where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra System to find $\frac { d R } { d T } .$
A)
$\frac { d R } { d T } = \frac { \sqrt { 0.004 T ^ { 3 } - 3 } } { 2 \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } }$
B)
$\frac { d R } { d T } = \frac { \sqrt { 0.004 T ^ { 3 } - 3 } } { 2 \sqrt { 0.001 T ^ { 4 } - 3 } }$
C)
$\frac { d R } { d T } = \frac { T ^ { 3 } } { 500 \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } }$
D)
$\frac { d R } { d T } = \frac { 0.004 T ^ { A } - 3 T } { 2 \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } }$
E)
$\frac { d R } { d T } = \frac { 0.004 T ^ { 3 } - 3 } { 2 \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } }$

Question

The resistance R of a certain type of resistor is $R = \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } ,$ where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra System to find $\frac { d R } { d T } .$ 
A)
$\frac { d R } { d T } = \frac { \sqrt { 0.004 T ^ { 3 } - 3 } } { 2 \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } }$
B)
$\frac { d R } { d T } = \frac { \sqrt { 0.004 T ^ { 3 } - 3 } } { 2 \sqrt { 0.001 T ^ { 4 } - 3 } }$
C)
$\frac { d R } { d T } = \frac { T ^ { 3 } } { 500 \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } }$
D)
$\frac { d R } { d T } = \frac { 0.004 T ^ { A } - 3 T } { 2 \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } }$
E)
$\frac { d R } { d T } = \frac { 0.004 T ^ { 3 } - 3 } { 2 \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } }$

Quizplus · Accepted Answer

Using the power rule and chain rule of differentiation, we have:

$$\frac { d R } { d T } = \frac {1}{2\sqrt{0.001T^4-3T+400}}\cdot \frac {d}{d T} \left(\sqrt{0.001T^4-3T+400}\, \right) $$
$$= \frac{1}{2 \sqrt{0.001T^4-3T+400}} \cdot \frac {1}{2\sqrt{0.001T^4-3T+400}} \cdot \frac{d}{d T} \left(0.001T^4-3T+400 \right) $$
$$= \frac {0.004T^3 - 3}{2\sqrt{0.001T^4-3T+400}}.$$

Therefore, the correct answer is (E).

The Resistance R of a Certain Type of Resistor Is R=0.001T4−3T+400,R = \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } ,R=0.001T4−3T+400​,

The Resistance R of a Certain Type of Resistor Is $R = \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } ,$