Chapter 2: Differentiation
Section 2.3: Differentiation Rules
Apply the rules in Table 2.3.1 to obtain the derivative of ft=t−1t2+2.
Apply the Quotient rule first. The derivatives needed in the course of applying the Quotient rule are obtained by applying the Sum, Difference, Constant, Identity, and Power rules.
Tools≻Load Package: Student Calculus 1
Context Panel: Assign Function
ft=t−1t2+2→assign as functionf
Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Simplify
f′t = 1t2+2−2⁢t−1⁢tt2+22= simplify −t2+2⁢t+2t2+22
Annotated stepwise solution
Expression palette: Differentiation template
Apply to ft.
Context Panel: Student Calculus1≻All Solution Steps
ⅆⅆ t⁡ft→show solution stepsDifferentiation Stepsⅆⅆtt−1t2+2▫1. Apply the quotient rule◦Recall the definition of the quotient ruleⅆⅆtt−1t2+2⁢g⁡t=ⅆⅆtt−1t2+2⁢g⁡t−x−1⁢ⅆⅆtg⁡tx2+2g⁡t2f⁡t=t−1g⁡t=t2+2This gives:ⅆⅆtt−1⁢t2+2−t−1⁢ⅆⅆtt2+2t2+22▫2. Apply the sum rule◦Recall the definition of the sum ruleⅆⅆtf⁡t+g⁡t=ⅆⅆtf⁡t+ⅆⅆtg⁡tf⁡t=tg⁡t=−1This gives:ⅆⅆtt+ⅆⅆt−1⁢t2+2−t−1⁢ⅆⅆtt2+2t2+22▫3. Apply the constant rule to the term ⅆⅆt−1◦Recall the definition of the constant ruleⅆⅆtC=0◦This meansⅆⅆt−1=0We can now rewrite the derivative as:ⅆⅆtt⁢t2+2−t−1⁢ⅆⅆtt2+2t2+22▫4. Apply the power rule to the term ⅆⅆtt◦Recall the definition of the power ruleⅆⅆtt=⁢t−1◦This means:ⅆⅆt=◦So,ⅆⅆtt=1We can rewrite the derivative as:t2+2−t−1⁢ⅆⅆtt2+2t2+22▫5. Apply the sum rule◦Recall the definition of the sum ruleⅆⅆtf⁡t+g⁡t=ⅆⅆtf⁡t+ⅆⅆtg⁡tf⁡t=t2g⁡t=2This gives:t2+2−t−1⁢ⅆⅆtt2+ⅆⅆt2t2+22▫6. Apply the constant rule to the term ⅆⅆt2◦Recall the definition of the constant ruleⅆⅆtC=0◦This meansⅆⅆt2=0We can now rewrite the derivative as:t2+2−t−1⁢ⅆⅆtt2t2+22▫7. Apply the power rule to the term ⅆⅆtt2◦Recall the definition of the power ruleⅆⅆtt=⁢t−1◦This means:ⅆⅆtt2=◦So,ⅆⅆtt2=We can rewrite the derivative as:t2+2−2⁢t−1⁢tt2+22
Alternatively, interactively apply the rules of differentiation via the Context Panel, as shown to the right.
Interactive solution by Differentiation Methods tutor
Figure 2.3.2(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, but the rules used are lost to the display.
The Close button returns an annotated version of the stepwise solution, similar in the form to the solution in Table 2.3.2(b).
Figure 2.3.2(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.
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