Use macros for standardized formatting of "derivative fractions"

chapters/functions.tex

@@ -1442,11 +1442,11 @@ \subsection{Using the Derivative Annotation}\label{using-the-derivative-annotati
 In the example above \lstinline!zeroDerivative=y! and \lstinline!zeroDerivative=offset! imply that
 \begin{equation*}
 \begin{aligned}
-\frac{d}{dt}f(x(t),y(t),o(t))
-&= \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial o}\frac{do}{dt}\\
-&= \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\cdot 0+\frac{\partial f}{\partial o}\cdot 0\\
-&= \frac{\partial f}{\partial x}\frac{dx}{dt}\\
-&= \mathit{fDer}\cdot\frac{dx}{dt}
+\udfrac{}{t}f(x(t),y(t),o(t))
+&= \pdfrac{f}{x} \udfrac{x}{t} + \pdfrac{f}{y} \udfrac{y}{t} + \pdfrac{f}{o} \udfrac{o}{t}\\
+&= \pdfrac{f}{x} \udfrac{x}{t} + \pdfrac{f}{y} \cdot 0 + \pdfrac{f}{o} \cdot 0\\
+&= \pdfrac{f}{x} \udfrac{x}{t}\\
+&= \mathit{fDer} \cdot \udfrac{x}{t}
 \end{aligned}
 \end{equation*}
 \end{nonnormative}
@@ -1498,11 +1498,11 @@ \subsection{Using the Derivative Annotation}\label{using-the-derivative-annotati
 Therefore \lstinline!h! indirectly includes the derivative with respect to \lstinline!y! as follows:
 \begin{equation*}
 \begin{aligned}
-\frac{d}{dt}\mathit{fg}(x(t))
-&= \frac{d}{dt}f(x(t),g(x(t)))\\
-&= \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\frac{dx}{dt}\\
-&= \left(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\right)\frac{dx}{dt}\\
-&= h(x,y)\frac{dx}{dt}
+\udfrac{}{t}\mathit{fg}(x(t))
+&= \udfrac{}{t}f(x(t),g(x(t)))\\
+&= \pdfrac{f}{x} \udfrac{x}{t} + \pdfrac{f}{y} \pdfrac{g}{x} \udfrac{x}{t}\\
+&= \left(\pdfrac{f}{x} + \pdfrac{f}{y} \pdfrac{g}{x} \right) \udfrac{x}{t}\\
+&= h(x,y) \udfrac{x}{t}
 \end{aligned}
 \end{equation*}
 \end{nonnormative}