Derive provides the menu items S
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
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
ariable command and specify
a nonnegative number. Next, we issue the M
command to control the direction of simplification and choose the
1: x :epsilon Real [0,inf) 2: Logarithm := ExpandThe expression
3: LN(x y)is now S
4: LN(x) + LN(y)The
Collectoption in the M
ogarithmmenu must be chosen to S
implifyin 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 Derive only does those simplifications about which it is sure that the transformations are valid, we enter under the above circumstances the following sum of logarithms.
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 M
rderingcommand the , , order of variables.
Trigonometric simplification is another highlight of Derive.
Use the M
rigonometry menu and choose
the direction of simplification and the preference of simplification
of powers of sines or cosines.
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 M
Toward: Cosinesand get
12: Trigpower := Cosines 2 13: 2 - COS(x)When you enter
14: SIN(x + y)and select M
15: Trigonometry := Expandthen simplification yields
16: COS(x) SIN(y) + SIN(x) COS(y)On its turn, choosing the
Collectdirection, gives back the original expression upon simplification.