Title: CR_Diff_01
Course: Demonstrating STACK (v4.3.7)

Derivatives of basic functions

Differentiate \([value]\) with respect to [value].
Feedback: Using the rule for differentiation \[\frac{d}{dx}(x^n)=nx^{n-1},\] \[ \frac{d}{d[value]}([value]) = [value][value]^{[value]-1} = [value][value]^[value].\]

Differentiate \([value]^[value]\) with respect to [value].
Feedback: Using the rule for differentiation \[\frac{d}{dx}(x^n)=nx^{n-1},\] \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}[[value]^[value]] &\displaystyle =&\displaystyle [value][value]^{[value]-1} \\ &\displaystyle =&\displaystyle [value][value]^[value]. \end{array}\]

Differentiate \([value]^[value]\) with respect to [value].
Feedback: Using the rule for differentiation \[\frac{d}{dx}(x^n)=nx^{n-1},\] \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}([value]) &\displaystyle =&\displaystyle [value][value]^{[value]-1} \\ &\displaystyle =&\displaystyle [value][value]^[value]. \end{array}\]

Differentiate \([value]\) with respect to [value].
Feedback: \[[value] = [value]^[value]\] So, using the rule for differentiation \[\frac{d}{dx}(x^n)=nx^{n-1},\] \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left([value]\right) &\displaystyle =&\displaystyle [value][value]^{[value]-1} \\ &\displaystyle =&\displaystyle [value][value]^[value]. \end{array}\]

Differentiate \[[value]\] with respect to [value]
Feedback: The derivative of \(\cos(x)\) with respect to \(x\) is \(-\sin(x)\), so \[\frac{d}{d[value]}\left[ [value]\right] =[value] .\]

Differentiate \[[value]\] with respect to [value]
Feedback: The derivative of \(\sin(x)\) with respect to \(x\) is \(\cos(x)\), so \[\frac{d}{d[value]}\left[ [value]\right] =[value][value] .\]

Differentiate \[[value]\] with respect to [value]
Feedback: The derivative of \(\tan(x)\) with respect to \(x\) is \(\sec^2(x)\), so \[\frac{d}{d[value]}\left[ [value]\right] =[value][value] .\]

Differentiate \[\sin([value])\] with respect to [value]
Feedback: The derivative of \(\sin(x)\) with respect to \(x\) is \(\cos(x)\), so \[\frac{d}{d[value]}\left[ [value]\right] =[value][value] .\]

Find the following derivative:
\[\frac{d}{d[value]}([value]).\]
Feedback: The derivative of \(e^z\) with respect to \(z\) is \(e^z\), so \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}([value]) &\displaystyle =&\displaystyle [value][value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following derivative: \[ \frac{d}{d[value]} [[value]]. \]
Feedback: The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\), so \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}[[value]] &\displaystyle =&\displaystyle [value]\cdot \frac{1}{[value][value]} \\ &\displaystyle =&\displaystyle \frac{1}[value] \end{array}\]

Find the following derivative: \[ \frac{d}{d[value]} [[value]]. \]
Feedback: The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\), so \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left[[value]\right] &\displaystyle =&\displaystyle [value]\cdot \frac{1}{[value][value]} \\ &\displaystyle =&\displaystyle \frac{1}[value] \end{array}\]