Title: CR_Diff_03
Course: Demonstrating STACK (v4.3.7)

Higher derivatives

Find the following second derivative: \[ \frac{d^2}{d[value]^2}([value]). \]
Feedback: The second derivative of a function \(f(x)\) with respect to \(x\) is \[\frac{d^2f}{dx^2}=\frac{d}{dx}\left(\frac{df}{dx}\right)\] So, if \(f([value])=[value]\) then its second derivative is \[\begin{array}{rcl}\displaystyle \frac{d^2}{d[value]^2}([value]) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[\frac{d}{d[value]}([value])\right] \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}([value]) \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following second derivative:
\[ \frac{d^2}{d[value]^2}([value]). \]
Feedback: The second derivative of a function \(f(x)\) with respect to \(x\) is \[\frac{d^2f}{dx^2}=\frac{d}{dx}\left(\frac{df}{dx}\right)\] So, if \(f([value])=[value]\) then its second derivative is \[\begin{array}{rcl}\displaystyle \frac{d^2}{d[value]^2}([value]) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[\frac{d}{d[value]}([value])\right] \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}([value]) \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following second derivative:
\[ \frac{d^2}{d[value]^2}\left[[value]\right]. \]
Feedback: The second derivative of a function \(f(x)\) with respect to \(x\) is \[\frac{d^2f}{dx^2}=\frac{d}{dx}\left(\frac{df}{dx}\right)\] So, if \(f([value])=[value]\) then its second derivative is \[\begin{array}{rcl}\displaystyle \frac{d^2}{d[value]^2}\left[([value]\right] &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[\frac{d}{d[value]}\left([value]\right)\right] \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[[value]\right] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following second derivative:
\[ \frac{d^2}{d[value]^2}([value]). \]
Feedback: The second derivative of a function \(f(x)\) with respect to \(x\) is \[\frac{d^2f}{dx^2}=\frac{d}{dx}\left(\frac{df}{dx}\right)\] So, if \(f([value])=[value]\) then its second derivative is \[\begin{array}{rcl}\displaystyle \frac{d^2}{d[value]^2}([value]) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[\frac{d}{d[value]}([value])\right] \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}\left([value]\right) \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following second derivative:
\[ \frac{d^2}{d[value]^2}\left([value]\right). \]
Feedback: The second derivative of a function \(f(x)\) with respect to \(x\) is \[\frac{d^2f}{dx^2}=\frac{d}{dx}\left(\frac{df}{dx}\right)\] So, if \(f([value])=[value]\) then its second derivative is \[\begin{array}{rcl}\displaystyle \frac{d^2}{d[value]^2}\left([value]\right) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[\frac{d}{d[value]}([value])\right] \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}\left([value]\right) \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Find the following second derivative: \[ \frac{d^2}{d[value]^2}\left([value]\right). \]
Feedback: The second derivative of a function \(f(x)\) with respect to \(x\) is \[\frac{d^2f}{dx^2}=\frac{d}{dx}\left(\frac{df}{dx}\right)\] So, if \(f([value])=[value]\) then its second derivative is \[\begin{array}{rcl}\displaystyle \frac{d^2}{d[value]^2}\left([value]\right) &\displaystyle =&\displaystyle \frac{d}{d[value]}\left[\frac{d}{d[value]}([value])\right] \\ &\displaystyle =&\displaystyle \frac{d}{d[value]}\left([value]\right) \\ &\displaystyle =&\displaystyle [value] \end{array}\]