`diff`

command is used to compute derivatives of Maple expressions.
Its syntax is basically
> diff(f,x);

where `f`

is an algebraic expression and `x`

is the variable
with respect to which the derivative is taken.
For example, to compute

,

we execute the Maple command
> diff((3*x-6)/(x^2-4),x);

You must specify the variable with respect to which you wish to take the derivative because, almost all the time, there are constants and other parameters around:

> diff(exp(a*x),x);

Maple knows theoretical results about derivatives. For example, even if
the functions `g(x)`

and `h(x)`

are not defined, Maple will give you
the derivative of their product in terms of the functions and their
derivatives:

> diff(h(x)*g(x),x);

> diff(diff(3*sin(x),x),x);

Such a command first calculates the derivative with respect to `x`

of the
expression `3*sin(x)`

and then differentiates the result to
obtain the second derivative. The same result can be accomplished with
either one of the following commands:

> diff(3*sin(x),x,x);

> diff(3*sin(x),x 2);

*Thereisanobviousextensionofthiscommandwhencalculatingmorethantwoderivatives*.*Youcanalsocalculatepartialderivativesofhigherorder* :

>

*diff* (*ln*(*x*)**exp*(- 2**x*), *x*3);

> diff(exp(x)*y^2*sin(z),x,y,z);

`y`

as a function of `x`

, you
must write `y(x)`

in your equation rather than just `y`

.
Here is a typical implicit differentiation problem. Consider the
equation
*x*^{2}*y* - 3*y*^{3}*x* = 0. Find the slope of the graph of the curve defined
by this equation at the point (3, 1):

> eq:=x^2*y(x)-3*y(x)^3*x = 0; deq:=diff(eq,x); solve(deq,diff(y(x),x));

> subs(y(x)=1,x=3,

The first command defines the equation relating `x`

and `y`

, the
second defines an equation `deq`

giving the relation between
`x`

, `y(x)`

and the derivative of `y`

with respect to
`x`

. Using `solve`

, we solve `deq`

to find the derivative as a
function of `x`

and `y(x)`

, and in the result we substitute
`x=3`

and `y=1`

to obtain the desired slope (see section 8
to learn about the `subs`

command).^{1.12}

Maple actually has a built-in command to do implicit differentiation,
`implicitdiff`

. Thus, we could have done this same problem more
concisely using the single command

> subs(y=1,x=3, implicitdiff(x^2y-3y^3x=0,y,x);

Occasionally, to make your worksheets easier to read, you may wish to have
Maple display a derivative in standard mathematical notation without
evaluating it. For this purpose there is an *inert* capitalized form
of the `diff`

command: `Diff`

. The two forms `diff`

and
`Diff`

are usually combined to produce meaningful sentences:

> Diff(exp(x)/(1-x),x);

> Diff(exp(x)/(1-x),x)=diff(exp(x)/(1-x),x);

This command is particularly useful to produce easy to read worksheets that involve partial derivatives:

> f:=x^3*exp(y)-sin(x*y);

> Diff(f,x,x,y)=diff(f,x,x,y);

- ... command).
^{1.12} - Again, the major caveat with the Maple derivative command is to avoid calculating derivatives with respect to a variable that has been previously given a value. In such an instance, unassign the value of the variable first.

2002-08-29