Differentiate an expression or a polynomial
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diff(f
) diff(f
,x
) diff(f, x1, x2, …
)
diff(f, x)
computes the derivative
of
the function f
with respect to the variable x
.
diff(f, x)
computes the partial derivative
of the arithmetical
expression (or polynomial) f
with respect
to the indeterminate x
.
diff(f)
computes the 0th derivative of f
.
Since the 0th derivative of f
is f
itself, diff(f)
returns
its evaluated argument.
diff(f, x1, x2, ...)
is equivalent to diff(...diff(diff(f,
x1), x2)...)
. In both cases, MuPAD^{®} first differentiates f
with
respect to x1
, then differentiates the result with
respect to x2
, and so on. The result is the partial
derivative
.
See Example 2.
If you use nested diff
calls, the system
internally converts them into a single diff
call
with multiple arguments. See Example 3.
When computing the second and higher derivatives, use the sequence operator as a shortcut. If n
is
a nonnegative integer, diff(f, x $ n)
returns the nth
derivative of f
with respect to x
.
See Example 4.
The indeterminates x, x1, x2, ...
must be
identifiers of domain type DOM_IDENT
or indexed
identifiers of the form x[n]
where x
is
an identifier and n
is an integer. If any indeterminate
comes in any other form, MuPAD returns an unresolved diff
call.
See Example 5.
If f
is an arithmetical expression, diff
returns
an arithmetical expression. If f
is a polynomial, diff
returns
a polynomial. See Example 6.
If the system cannot compute the derivative, it returns an unresolved diff
call.
See Example 7.
MuPAD assumes that partial derivatives with respect to
different indeterminates commute. The function calls diff(f,
x1, x2)
and diff(f, x2, x1)
produce the
same result diff(f, y1, y2)
. Here [y1,
y2]
= sort([x1, x2])
. See Example 8.
MuPAD provides two functions, diff
and D
, for computing
derivatives. Use the differential operator D
to compute the derivatives
of functions.
Use the diff
function to compute the derivatives
of arithmetical
expressions. Mathematically, D(f)(x)
coincides
with diff(f(x), x)
and D([1, 2], f)(x,
y)
coincides with diff(f(x, y), x, y)
.
You can convert symbolic calls of D
to the calls of diff
and
vice versa by using rewrite
.
See Example 10.
You can extend the functionality of diff
for
your own special mathematical functions via overloading.
This approach works by turning the corresponding function into a function environment and implementing the
differentiation rule for the function as the "diff"
slot
of the function
environment.
If a subexpression of the form g(..)
occurs
in f
, and g
is a function environment, then diff(f,
x)
attempts to call the "diff"
slot of g
to
determine the derivative of g(..)
.
The system calls the "diff"
slot with the
arguments g(..), x
.
If g
does not have a "diff"
slot,
then the system function diff
returns the symbolic
expression diff(g(..), x)
for the derivative of
the subexpression.
The system always calls the "diff"
slot with
exactly two arguments. If you call the diff
function
with more indeterminates (for example, if you compute a higher derivative),
then MuPAD calls the "diff"
slot several times.
Each call computes the derivative with respect to one indeterminate.
The system caches the results of the calls of "diff"
slots
in diff
in order to prevent redundant function
calls. See Example 11.
Similarly, if an element d
of a library domain T
occurs
as a subexpression of f
, then diff(f,
x)
calls the slot T::diff(d, x)
to compute
the derivative of d
.
If the domain T
does not have a "diff"
slot,
then diff
considers this object as a constant and
returns 0 for the corresponding subexpression.
If you differentiate an expression or function containing abs
or sign
,
ensure that the arguments are real values. For complex arguments of abs
and sign
,
the diff
function formally computes the derivative,
but this result is not generally valid because abs
and sign
are
not differentiable over complex numbers.
Compute the derivative of x^{2} with
respect to x
:
diff(x^2, x)
You can differentiate with respect to multiple variables within
a single diff
call. For example, differentiate
this expression with respect to x
, and then with
differentiate the result with respect to y
:
diff(x^2*sin(y), x, y) = diff(diff(x^2*sin(y), x), y)
MuPAD internally converts nested diff
calls
into a single diff
call with multiple arguments:
diff(diff(f(x, y), x), y)
Use the sequence operator $
as a shortcut to compute the third
derivative of this expression with respect to x
:
diff(sin(x)*cos(x), x $ 3)
You can differentiate with respect to an indexed identifier.
For example, differentiate this expression with respect to x[1]
:
diff(x[1]*y + x[1]*x[r], x[1])
You can differentiate polynomials
with respect to the polynomial
indeterminates or the parameters in the coefficients. For example,
differentiate this polynomial with respect to the indeterminate x
:
diff(poly(sin(a)*x^3 + 2*x, [x]), x)
Now differentiate the same polynomial with respect to its symbolic
parameter a
:
diff(poly(sin(a)*x^3 + 2*x, [x]), a)
MuPAD returns the derivative of an unknown function as
an unresolved diff
call:
diff(f(x) + x, x)
MuPAD assumes that all partial derivatives with respect to different indeterminates commute. Therefore, the system can change the order of indeterminates:
diff(f(x, y), x, y) = diff(f(x, y), y, x);
You can use diff
to differentiate symbolic
integrals. For example, compute the second derivative of this indefinite
integral:
F1 := int(f(x), x): diff(F1, x, x)
Now compute the derivative of the definite integral:
F2 := int(f(t, x), t = x..x^2): diff(F2, x)
Use the operator D
to compute the derivatives of functions.
Use the diff
function to compute the derivatives
of expressions:
D(sin), diff(sin(x), x)
Applying D
to
expressions and diff
to functions makes no sense:
D(sin(x)), diff(sin, x)
Use the rewrite
function
to rewrite an expression replacing the operator D
with
the diff
function:
rewrite(D(f)(x), diff), rewrite(D(D(f))(x), diff)
Also, use rewrite
to
rewrite an expression replacing diff
with D
:
diff(f(x, x), x) = rewrite(diff(f(x, x), x), D)
You can extend diff
to your own special functions.
To do so, embed your function, f
, into a function environment, g
,
and implement the behavior of diff
for this function
as the "diff"
slot of
the function environment.
If a subexpression of the form g(..)
occurs
in an expression f
, then diff(f, x)
calls g::diff(g(..),
x)
to determine the derivative of the subexpression g(..)
.
This example demonstrates extending diff
to
the exponential function. Since the function environment exp
already has a "diff"
slot,
call the new function environment Exp
to avoid
overwriting the existing system function exp
.
Here, the "diff"
slot implements the chain
rule for the exponential function. The derivative is the product of
the original function call and the derivative of the argument:
Exp := funcenv(Exp): Exp::diff := proc(f, x) begin // f = Exp(something), i.e., something = op(f, 1) f*diff(op(f, 1), x): end_proc: diff(Exp(x^2), x)
The report created by prog::trace
shows
one call to Exp::diff
with two arguments. Instead
of calling Exp::diff
twice, the system reads the
required result of the second call from an internal cache for intermediate
results in diff
:
prog::trace(Exp::diff): diff(Exp(x^2), x, x)
enter Exp::diff(Exp(x^2), x) computed 2*x*Exp(x^2)
prog::untrace(Exp::diff): delete f, Exp:

An arithmetical
expression or a polynomial of type 

Indeterminates: identifiers or indexed identifiers 
f