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+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Differentiation
+
+%  Author: Clifton J. Williamson
+%  Date created: 1 November 1989
+%  Date last updated: 1 November 1989
+
+\begin{page}{ExDiffBasic}{Computing Derivatives}
+\beginscroll
+To compute a derivative, you must specify an expression and a variable
+of differentiation.
+For example, to compute the derivative of {\em sin(x) * exp(x**2)} with respect to the
+variable {\em x}, issue the following command:
+\spadpaste{differentiate(sin(x) * exp(x**2),x)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffSeveralVariables}{Derivatives of Functions of Several Variables}
+\beginscroll
+Partial derivatives are computed in the same way as derivatives of functions
+of one variable: you specify the function and a variable of differentiation.
+For example:
+\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)}
+\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffHigherOrder}{Derivatives of Higher Order}
+\beginscroll
+To compute a derivative of higher order (e.g. a second or third derivative),
+pass the order as the third argument of the function 'differentiate'.
+For example, to compute the fourth derivative of {\em exp(x**2)}, issue the
+following command:
+\spadpaste{differentiate(exp(x**2),x,4)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffMultipleI}{Multiple Derivatives I}
+\beginscroll
+When given a function of several variables, you may take derivatives repeatedly
+and with respect to different variables.
+The following command differentiates the function {\em sin(x)/(x**2 + y**2)}
+first with respect to {\em x} and then with respect to {\em y}:
+\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y])}
+As you can see, we first specify the function and then a list of the variables
+of differentiation.
+Variables may appear on the list more than once.
+For example, the following command differentiates the same function with
+respect to {\em x} and then twice with respect to {\em y}.
+\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y,y])}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffMultipleII}{Multiple Derivatives II}
+\beginscroll
+You may also compute multiple derivatives by specifying a list of variables
+together with a list of multiplicities.
+For example, to differentiate {\em cos(z)/(x**2 + y**3)}
+first with respect to {\em x},
+then twice with respect to {\em y}, then three times with respect to {\em z},
+issue the following command:
+\spadpaste{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffFormalIntegral}{Derivatives of Functions Involving Formal Integrals}
+\beginscroll
+When a function does not have a closed-form antiderivative, \Language{}
+returns a formal integral.
+A typical example is
+\spadpaste{f := integrate(sqrt(1 + t**3),t) \bound{f}}
+This formal integral may be differentiated, either by itself or in any
+combination with other functions:
+\spadpaste{differentiate(f,t) \free{f}}
+\spadpaste{differentiate(f * t**2,t) \free{f}}
+\endscroll
+\autobuttons\end{page}