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ExtraStep
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HansOlsson committed Feb 19, 2021
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3 changes: 2 additions & 1 deletion chapters/functions.tex
Expand Up @@ -1468,7 +1468,8 @@ \subsection{Using the Derivative Annotation}\label{using-the-derivative-annotati

Therefore \lstinline!f_der! indirectly includes the derivative with respect to \lstinline!y! as follows:
\begin{eqnarray*}
\frac{d}{dt}fg(x(t))&=&\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\frac{dx}{dt}\\
\frac{d}{dt}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}\\
&=&(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial x})\frac{dx}{dt}\\
&=&\textrm{f\_der}(x,y)\frac{dx}{dt}
\end{eqnarray*}
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