Title: CR_Diff_05
Course: Demonstrating STACK (v4.4.7)

Quotient rule

Differentiate \[[value]\] with respect to [value].
Feedback: In order to solve this problem we will have to use the quotient rule calc_quotient_rule as follows: \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left([value]\right) &\displaystyle =&\displaystyle \frac{\left[\frac{d}{d[value]}([value])\right]\cdot ([value]) - ([value])\cdot \left[\frac{d}{d[value]}([value])\right]}{([value])^2} \\ &\displaystyle =&\displaystyle \frac{[[value]]\cdot ([value]) - ([value])\cdot [[value]]}{([value])^2} \\ &\displaystyle =&\displaystyle \frac{[value] - [value]}{([value])^2} \\ &\displaystyle =&\displaystyle \frac[value]{([value])^2} \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Differentiate \[[value]\] with respect to [value].
Feedback: In order to solve this problem we will have to use the quotient rule calc_quotient_rule as follows: \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left([value]\right) &\displaystyle =&\displaystyle \frac{\left[\frac{d}{d[value]}([value])\right]\cdot ([value]) - ([value])\cdot \left[\frac{d}{d[value]}([value])\right]}{([value])^2} \\ &\displaystyle =&\displaystyle \frac{[[value])]\cdot ([value]) - ([value])\cdot [[value]]}{([value])^2} \\ &\displaystyle =&\displaystyle \frac{[value] - [value]}{([value])^2} \\ &\displaystyle =&\displaystyle \frac[value]{([value])^2} \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following derivative:
\[ \frac{d}{d[value]}\left[[value]\right]. \]
Feedback: In order to solve this problem we will have to use the quotient rule calc_quotient_rule as follows: \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[[value]\right] &\displaystyle =&\displaystyle \frac{\left[\frac{d}{d[value]}([value])\right]\cdot [value] - [value]\cdot \left[\frac{d}{d[value]}[[value]]\right]}{[[value]]^2} \\ &\displaystyle =&\displaystyle \frac{[[value]]\cdot [value] - [value]\cdot [[value]]}[value] \\ &\displaystyle =&\displaystyle \frac{[value] - [value]}[value] \end{array}\]

Find the following derivative:
\[ \frac{d}{d[value]}\left[[value]\right]. \]
Feedback: In order to solve this problem we will have to use the quotient rule calc_quotient_rule as follows: \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[[value]\right] &\displaystyle =&\displaystyle \frac{\left[\frac{d}{d[value]}[[value]]\right]\cdot [value] - [value]\cdot \left[\frac{d}{d[value]}[[value]]\right]}{[[value]]^2} \\ &\displaystyle =&\displaystyle \frac{\left[[value]\right]\cdot [value] - [value]\cdot \left[[value]\right]}[value] \\ &\displaystyle =&\displaystyle \frac{[value] - [value]}[value] \\ &\displaystyle =&\displaystyle \frac[value][value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following derivative: \[ \frac{d}{d[value]}\left[[value]\right]. \]
Feedback: In order to solve this problem we will have to use the quotient rule calc_quotient_rule as follows: \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[[value]\right] &\displaystyle =&\displaystyle \frac{\left[\frac{d}{d[value]}([value])\right]\cdot [value] - [value]\cdot \left[\frac{d}{d[value]}[[value]]\right]}{[[value]]^2} \\ &\displaystyle =&\displaystyle \frac{[[value]]\cdot [value] - [value]\cdot [[value]]}[value] \\ &\displaystyle =&\displaystyle \frac{[value] - [value]}[value] \end{array}\]

Find the following derivative:
\[ \frac{d}{d[value]}\left[[value]\right]. \]
Feedback: In order to solve this problem we will have to use the quotient rule calc_quotient_rule as follows: \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[[value]\right] &\displaystyle =&\displaystyle \frac{\left[\frac{d}{d[value]}([value])\right]\cdot [value] - [value]\cdot \left[\frac{d}{d[value]}[[value]]\right]}{[[value]]^2} \\ &\displaystyle =&\displaystyle \frac{[[value]]\cdot [value] - [value]\cdot [[value]]}[value] \\ &\displaystyle =&\displaystyle \frac{[value] - [value]}[value]\\ &\displaystyle =&\displaystyle [value]. \end{array}\]