Differentiation Rules for Calculus1
Rules
Examples
See Student[Calculus1] for a general introduction to the Calculus1 subpackage of the Student package.
See SingleStepOverview for an introduction to the step-by-step (or single-step) functionality of the Calculus1 package.
The following table lists the built-in rules for differentiation that do not take parameters. These rules can be passed as the index to Rule or as a rule argument to Understand.
Rule
Alternate Names
Description
chain
fgx′=f′gxg′x
constant
c′=0
constantmultiple
`c*`
cf′=cf′
difference
`-`
f−g′=f′−g′
identity
`^`
x′=1
int
Int
∫cxftⅆt' =fx
power
xn′=nxn−1
product
`*`
fg′=f′g+fg′
quotient
`/`
fg′=gf′−fg′g2
sum
`+`
f+g′=f′+g′
The name of any univariate function can also be used as a rule argument to the Rule command. The name of any univariate function recognized by Maple, for example, sin, can be passed as a rule argument to the Understand command (where recognized means that it is of type mathfunc).
There is one differentiation rule which requires a parameter: rewrite. This rule can be used as the index to a call to Rule, but cannot be given as a rule argument to Understand. This rule is used to change the form of the expression being differentiated. It has the general form:
[rewrite, f1x=g1x, f2x=g2x, ...]
The effect of applying the rewrite rule is to perform each substitution listed as a parameter to the rule, where occurrences of the left-hand side of each substitution are replaced by the corresponding right-hand side.
The main application of this rule is to rewrite an expression of the form fxgx, where the exponent (at least) depends on the differentiation variable, as an exponential. The rule would thus be given as:
[rewrite, fxgx=ⅇgxlnfx ]
Note: The Rule routine does not attempt to validate the rewrite rules you provide.
withStudent:-Calculus1:
infolevelStudentCalculus1≔1:
Rule`*`Diffx2sinx2,x
Creating problem #1
ⅆⅆxx2sinx2=ⅆⅆxx2sinx2+x2ⅆⅆxsinx2
Rulechain
ⅆⅆxx2sinx2=ⅆⅆxx2sinx2+x2ⅆⅆ_X0sin_X0_X0=x2|ⅆⅆ_X0sin_X0_X0=x2ⅆⅆxx2
Rulesin
ⅆⅆxx2sinx2=ⅆⅆxx2sinx2+x2cosx2ⅆⅆxx2
If the operation type is ambiguous, Maple returns an error
RulesumDiffx2+Intcost,t=0..x,x
Error, (in Student:-Calculus1:-Rule[sum]) unable to determine which calculus operation is being applied in this problem; you can provide this information as the 2nd argument on your call to Rule or Hint
RulesumDiffx2+Intcost,t=0..x,x,diff
Creating problem #2
∂∂xx2+∫0xcostⅆt=ⅆⅆxx2+∂∂x∫0xcostⅆt
Ruleint
∂∂xx2+∫0xcostⅆt=ⅆⅆxx2+cosx
Rule`^`Diffexpx,x
Creating problem #3 Rule [power] does not apply
ⅆⅆxⅇx=ⅆⅆxⅇx
Rulerewrite,xsinx=expsinxlnxDiffxsinx,x
Creating problem #4
ⅆⅆxxsinx=ⅆⅆxⅇsinxlnx
This example illustrates how to handle an unknown univariate function.
Rule`*`Diffrfr,r
Creating problem #5
ⅆⅆrrfr=ⅆrⅆrfr+rⅆⅆrfr
Rulef
Ruleidentity
ⅆⅆrrfr=fr+rⅆⅆrfr
ShowIncomplete
The current problem is complete
See Also
diff
Diff
Student
Student[Calculus1]
Student[Calculus1][DiffTutor]
Student[Calculus1][SingleStepOverview]
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