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Doing Calculus with MATLAB 61

Doing Calculus with MATLAB

MATLAB has commands for most of the computations of basic calculus
in its Symbolic MathToolbox. This toolbox includes part of a separate program

called Maple , which processes the symbolic calculations.

Differentiation

You can use diff to differentiate symbolic expressions, and also to approxi-
mate the derivative of a function given numerically (say by an M-file):

>> syms x; diff(xˆ3)

ans =
3*x^2

Here MATLAB has figured out that the variable is x. (See Default Variables
at the end of the chapter.) Alternatively,

>> f = inline(’xˆ3’, ’x’); diff(f(x))

ans =
3*x^2

The syntax for second derivatives is diff(f(x), 2), and for nthderivatives,
diff(f(x), n). The command diff can also compute partial derivatives
of expressions involving several variables, as in diff(xˆ2*y, y), but to do
multiple partials withrespect to mixed variables you must use diff repeat-
edly, as in diff(diff(sin(x*y/z), x), y). (Remember to declare y and
z symbolic.)
There is one instance where differentiation must be represented by the
letter D, namely when you need to specify a differential equation as input to
a command. For example, to use the symbolic ODE solver on the differential

equation xy + 1 = y, you enter
dsolve(’x*Dy + 1 = y’, ’x’)