*Derive* provides the menu items **S**`implify`

, **E**`xpand`

,
and **F**`actor`

to simplify expressions in which special functions
like trigonometric functions, logarithm, and exponential occur.
*Derive* is very precise in checking whether transformations of formulae are
valid or not on the domain of computation. If necessary, you have to issue
the **D**`eclare`

**V**`ariable`

command and restrict the domain
of a variable. For functions with simplification rules in two directions such
as the rule you also have to specify the direction
via the **M**`anage`

menu item. Examples show best what can be achieved.

Let us consider the transformation in both directions.
It is only valid when is real or complex and is nonnegative (or with
the roles of and interchanged). *Derive* refuses to do the transformation
unless the necessary condition is satisfied. So, let us first issue
the **D**`eclare`

**V**`ariable`

command and specify `x`

as
a nonnegative number. Next, we issue the **M**`anage`

**L**`ogarithm`

command to control the direction of simplification and choose the `Expand`

option.

1: x :epsilon Real [0,inf) 2: Logarithm := ExpandThe expression

3: LN(x y)is now

`implified`

to
4: LN(x) + LN(y)The

`Collect`

option in the `anage`

`ogarithm`

menu
must be chosen to `implify`

in the opposite direction and combine
sum of logarithms into the logarithm of a product.
5: Logarithm := Collect 6: LN(x y)In order to illustrate that

7: LN(x) + LN(y) + LN(z)Simplification only partly combines logarithms.

8: LN(x y) + LN(z)You can get the simplification

9: LN(x z) + LN(y)by choosing in the

`anage`

`rdering`

command the
, , order of variables.
Trigonometric simplification is another highlight of *Derive*.
Use the **M**`anage`

**T**`rigonometry`

menu and choose
the direction of simplification and the preference of simplification
of powers of sines or cosines.
The `Auto`

option can be chosen when you leave it up to *Derive* to
make a heuristic choice.
For example, in the `Auto`

mode, *Derive* simplifies

2 2 10: COS(x) + 2 SIN(x)into

2 11: SIN(x) + 1But if you prefer cosines over sines, you select

`anage`

`rigonometry`

`Toward: Cosines`

and get
12: Trigpower := Cosines 2 13: 2 - COS(x)When you enter

14: SIN(x + y)and select

`anage`

`rigonometry`

`Direction: Expand`

,
15: Trigonometry := Expandthen simplification yields

16: COS(x) SIN(y) + SIN(x) COS(y)On its turn, choosing the

`Collect`

direction, gives back the original
expression upon simplification.

Thu Mar 23 17:40:24 MET 1995