Chapter 2: Differentiation
Section 2.3: Differentiation Rules
Apply the rules in Table 2.3.1 to obtain the derivative of fx=x3−3⁢x2+x+3.
Tools≻Load Package: Student Calculus1
Context Panel: Assign Function
fx=x3−3⁢x2+x+3→assign as functionf
A Stepwise Solution
Table 2.3.1(a) summarizes the steps that might be used to evaluate the derivative of x3−3⁢x2+x+3.
applied Sum and Difference rules
applied Constant rule
applied Constant Multiple rule
applied Identity and Power rules
arithmetic ⇒ final answer
Table 2.3.1(a) Differentiation of fx=x3−3⁢x2+x+3
A Maple-Generated Stepwise Solution
Table 2.3.1(b) explains how to obtain an annotated stepwise solution in Maple.
Calculus palette: Differentiation template
Apply to fx.
Context Panel: Student Calculus1≻All Solution Steps
ⅆⅆ x⁡x3−3 x2+x+3→show solution stepsDifferentiation Stepsⅆⅆxx3−3⁢x2+x+3▫1. Apply the sum rule◦Recall the definition of the sum ruleⅆⅆxf1⁡x+f2⁡x+f3⁡x+f4⁡x=ⅆⅆxf1⁡x+ⅆⅆxf2⁡x+ⅆⅆxf3⁡x+ⅆⅆxf4⁡xf1⁡x=x3f2⁡x=−3⁢x2f3⁡x=xf4⁡x=3This gives:ⅆⅆxx3+ⅆⅆx−3⁢x2+ⅆⅆxx+ⅆⅆx3▫2. Apply the constant rule to the term ⅆⅆx3◦Recall the definition of the constant ruleⅆⅆxC=0◦This meansⅆⅆx3=0We can now rewrite the derivative as:ⅆⅆxx3+ⅆⅆx−3⁢x2+ⅆⅆxx▫3. Apply the constant multiple rule to the term ⅆⅆx−3⁢x2◦Recall the definition of the constant multiple ruleⅆⅆx⁢f⁡x=⁢ⅆⅆxf⁡x◦This means:ⅆⅆx−3⁢x2=We can rewrite the derivative as:ⅆⅆxx3−3⁢ⅆⅆxx2+ⅆⅆxx▫4. Apply the power rule to the term ⅆⅆxx2◦Recall the definition of the power ruleⅆⅆxx=⁢x−1◦This means:ⅆⅆxx2=◦So,ⅆⅆxx2=We can rewrite the derivative as:ⅆⅆxx3−6⁢x+ⅆⅆxx▫5. Apply the power rule to the term ⅆⅆxx◦Recall the definition of the power ruleⅆⅆxx=⁢x−1◦This means:ⅆⅆx=◦So,ⅆⅆxx=1We can rewrite the derivative as:ⅆⅆxx3−6⁢x+1▫6. Apply the power rule to the term ⅆⅆxx3◦Recall the definition of the power ruleⅆⅆxx=⁢x−1◦This means:ⅆⅆxx3=We can rewrite the derivative as:3⁢x2−6⁢x+1
Table 2.3.1(b) Annotated stepwise solution generated in Maple
Interactive Stepwise Solution
Context Panel: Student Calculus1≻Apply Next Differentiation Step≻[sum] (See Figure 2.3.1(a).)
Context Panel: Student Calculus1≻Differentiation Rules≻sum (See Figure 2.3.1(b).)
Figure 2.3.1(a) Maple-provided differentiation rule
Figure 2.3.1(b) User-selected differentiation rule
Solution via Differentiation Methods Tutor
Figure 2.3.1(a) shows the
tutor applied to the given expression. To launch this tutor with the expression already embedded, first load the Student Calculus1 package, then from the Context Panel, select Tutors and the particular tutor to be launched.
Apply the rules of differentiation by clicking the corresponding button in the tutor.
The Next Step button will provide one step in the solution; the All Steps button will display all the steps of the calculation.
The Close button returns an annotated version of the stepwise solution, similar in the form to the solution in Table 2.3.1(b).
Figure 2.3.1(a) Differentiation Methods tutor
The menu bar provides a summary of each known rule (Rule Definition), Help, and another way to apply rules (Apply Rule). Note that the selected rule is generally applied to the first possible occurrence in the expression; it may be necessary to apply a rule multiple times in succession. Rules that are thoroughly understood can be marked as "understood" (via Understood Rules) so that their application becomes automatic
If the tutor is launched from the Tools≻Tutors menu, the function to be differentiated has to be entered, either by typing, or by copy/paste.
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