Title: CR_Diff_02
Course: Demonstrating STACK (v4.4.7)

Linearity in diﬀerentiation

Differentiate \[[value]\] with respect to [value].
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[ [value]\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[[value]\right] +\frac{d}{d[value]}\left([value]\right) \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}[[value]]+\frac{d}{d[value]}\left([value]\right) \\ &\displaystyle =&\displaystyle [value]\cdot [[value]]+[value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Differentiate \[[value]\] with respect to [value].
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[ [value]\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[[value]\right] +\frac{d}{d[value]}\left( [value]\right) \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}[[value]]+[value]\cdot \frac{d}{d[value]}\left( [value]\right) \\ &\displaystyle =&\displaystyle [value]\cdot [[value]]+[value]\cdot \left[ [value]\right] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Differentiate \[[value]\] with respect to [value].
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[ [value]\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[[value]\right] +\frac{d}{d[value]}\left[[value]\right]+\frac{d}{d[value]}\left[[value]\right] \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}[[value]]+\frac{d}{d[value]}\left[[value]\right] +[value]\cdot \frac{d}{d[value]}\left[[value]\right] \\ &\displaystyle =&\displaystyle [value]+[[value]]+[value]\cdot \left[[value]\right] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Differentiate \[[value]\] with respect to [value].
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[ [value]\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[[value]\right] +\frac{d}{d[value]}\left[[value]\right]+\frac{d}{d[value]}\left[[value]\right] \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}[[value]]+[value]\cdot \frac{d}{d[value]}\left[[value]\right] +[value]\cdot \frac{d}{d[value]}\left[[value]\right] \\ &\displaystyle =&\displaystyle [value]\cdot [[value]]+[value]\cdot [[value]] +[value]\cdot \left[[value]\right] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following derivative:
\[\frac{d}{d[value]}\left( [value]\right) .\]
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left( [value]\right) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left( [value]\right) +\frac{d}{d[value]}\left[ [value]\right] \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}([value])+[value]\cdot \frac{d}{d[value]}[[value]] \\ &\displaystyle =&\displaystyle [value]\cdot \left( [value]\right) +[value]\cdot \left[ [value]\right] \\ &\displaystyle =&\displaystyle [value]+[value] \end{array}\]

Find the following derivative:
\[\frac{d}{d[value]}\left( [value]\right) .\]
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. [[facts:calc_diff_linearity_rule]] \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left( [value]\right) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left( [value]\right) +\frac{d}{d[value]}\left[ [value]\right] \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}([value])+[value]\cdot \frac{d}{d[value]}[[value]] \\ &\displaystyle =&\displaystyle [value]\cdot \left( [value]\right) +[value]\cdot \left[ [value]\right] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following derivative:
\[\frac{d}{d[value]}\left( [value]\right) .\]
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left( [value]\right) &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}([value]) \\ &\displaystyle =&\displaystyle [value]\cdot \left( [value]\right) \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following derivative:
\[\frac{d}{d[value]}\left( [value]\right) .\]
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left( [value]\right) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left( [value]\right) +\frac{d}{d[value]}\left[ [value]\right] \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}([value])+[value]\cdot \frac{d}{d[value]}[[value]] \\ &\displaystyle =&\displaystyle [value]\cdot \left( [value]\right) +[value]\cdot \left[ [value]\right] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following derivative: \[\frac{d}{d[value]}\left( [value]\right) .\]
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left( [value]\right) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left( [value]\right) +\frac{d}{d[value]}\left[ [value]\right] \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}([value])+[value]\cdot \frac{d}{d[value]}[[value]] \\ &\displaystyle =&\displaystyle [value]\cdot \left( [value]\right) +[value]\cdot \left[ [value]\right] \\ &\displaystyle =&\displaystyle [value]+[value] \end{array}\]

Expand the powers and find the following derivative:
\[ \frac{d}{d[value]}\left[\sqrt[value]\right] . \]
Feedback: In order to solve this problem we will have to use the linearity rules for differentition calc_diff_linearity_rule and the properties of exponentials alg_indices. \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[\sqrt{\left([value]\right)}\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left( [value]\right) \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}([value]) \\ &\displaystyle =&\displaystyle [value]\cdot [value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Expand the powers and find the following derivative:
\[ \frac{d}{d[value]}\left[\left([value]\right)^[value]\right] . \]
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[\left([value]\right)^[value]\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left( [value]\right) \\ &\displaystyle =&\displaystyle [value]\cdot \frac{d}{d[value]}([value]) \\ &\displaystyle =&\displaystyle [value]\cdot [value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Expand the following expression and differentiate it with respect to [value]:\[ [value]. \]
Feedback: We can expand out \[ [value]=[value]. \] Now it is relatively easy to differentiate term by term to get \[ [value]. \]

Expand the following expresion and differentiate it with respect to [value]:\[ \frac[value][value]. \]
Feedback: In order to solve this problem we will have to use the linearity rules for differentition. calc_diff_linearity_rule Since \[\begin{array}{rcl}\displaystyle \frac[value][value] &\displaystyle =&\displaystyle \frac{1}[value]-\frac[value][value] \\ &\displaystyle =&\displaystyle [value] \end{array}\] \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[\frac[value][value]\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left( [value]\right) \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}\left([value]\right) - \frac{d}{d[value]}\left( [value]\right) \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Use the laws of logarithms to find the following derivative: \[ \frac{d}{d[value]} \left[\ln([value])\right]. \]
Feedback: Using the laws of logarithms alg_logarithms we get \[\begin{array}{rcl}\displaystyle \ln([value]) &\displaystyle =&\displaystyle [value][value] \end{array}\] So, by linearity calc_diff_linearity_rule, \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[\ln([value])\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[[value][value]\right] \\ &\displaystyle =&\displaystyle [value]\frac{d}{dx}[[value]] \\ &\displaystyle =&\displaystyle [value][value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Use the laws of logarithms to find the following derivative: \[ \frac{d}{d[value]} \left[[value]\right]. \]
Feedback: Using the laws of logarithms alg_logarithms we get \[\begin{array}{rcl}\displaystyle [value] &\displaystyle =&\displaystyle \ln(1) - \ln([value]) \\ &\displaystyle =&\displaystyle -\ln([value]) - \ln([value]) \\ &\displaystyle =&\displaystyle [value][value] - \ln([value]) \end{array}\] So, by linearity calc_diff_linearity_rule, \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[[value]\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[[value][value] - \ln([value])\right] \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[[value][value]\right] + \frac{d}{d[value]}\left[-\ln([value])\right] \\ &\displaystyle =&\displaystyle [value]\frac{d}{d[value]}\left[[value]\right] + 0 \\ &\displaystyle =&\displaystyle [value][value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]