To compute derivatives of functions, use the differential operator D
. This operator
differentiates both standard mathematical functions and your own functions
created in MuPAD^{®}. For example, find the first derivatives of
the following standard mathematical functions implemented in MuPAD:
D(sin), D(exp), D(cosh), D(sqrt), D(heaviside)
Create your own function with one variable and compute a derivative of this function:
f := x -> x^3: D(f)
f := x -> sin(x)/x^2: f'; f'(x)
Computing the first derivatives of a function lets you find
its local extrema (minima and maxima). For example, create this function
and plot it on the interval -10 < x < 10
:
F := x -> x^3*sin(x); plot(F, x = -10..10)
Find the local extrema of F
on the interval -10
< x < 10
. If the point is a local extremum (either
minimum or maximum), the first derivative of the function at that
point equals 0. Therefore, to find the local extrema of F
,
solve the equation F'(x) = 0
. Use the AllRealRoots
option
to return more than one solution.
extrema := numeric::solve(F'(x) = 0, x = -10..10, AllRealRoots)
Now, compute the corresponding values of F
.
For example, compute F
for the third element, -2.455643863
,
in the solution set:
F(extrema[3])
To compute the values of F
for all local
minima and maxima, use the following command. Here, $
is
used to evaluate F
for every element of the extrema
set.
points := {[x, F(x)] $ x in extrema}
Plot function F
with extrema points:
plot(F, points, x = -10..10)
To compute a derivative of a multivariable function, specify
the differentiation variable. The operator D
does not accept the
variable names. Instead of providing a variable name, provide its
index. For example, integrate the following function with respect
to its first variable x
. Then integrate the function
with respect to its second variable y
:
f := (x, y) -> x^2 + y^3: D([1], f); D([2], f)
The list of indices accepted by the operator D
refers to the order
in which you provided the variables when creating a function:
f := (x, y) -> x^2 + y^3: D([1], f); f := (y, x) -> x^2 + y^3: D([1], f)
To find second- and higher-order partial derivatives of a function,
use the same index two or more times. For example, compute the second-order
partial derivatives with respect to x
and with
respect to y
:
f := (x, y) -> x^3*sin(y): D([1, 1], f); D([2, 2], f)
To compute second- and higher-order derivatives with respect to several variables (mixed derivatives), provide a list of indices of differentiation variables:
f := (x, y) -> x^3*sin(y): D([1, 2], f);
Note: To improve performance, MuPAD assumes that all mixed derivatives commute. For example, . This assumption suffices for most engineering and scientific problems. |