Chapter 2: Differentiation
Section 2.2: Precise Definition of the Derivative
Example 2.2.3
Apply Definition 2.2.1 to fx=x−1, to obtain f′c, c>1.
Solution
Define the function f
Control-drag fx=… Context Panel: Assign Function
fx=x−1→assign as functionf
Method 1
Write the mathematical notation for the derivative
Context Panel: Evaluate and Display Inline
f′c = 12c−1
Method 2
Type fx and press the Enter key.
Context Panel: Differentiate≻With Respect To≻x
Context Panel: Evaluate at a Point≻x=c
fx
x−1
→differentiate w.r.t. x
12x−1
→evaluate at point
12c−1
Method 3
Expression palette: Differentiation template Apply to fx and press the Enter key.
Evaluate at x=c as above.
ⅆⅆ x fx
Stepwise solutions
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
Apply the NewtonQuotient command.
Context Panel: Limit Select h as the variable Set h=0
NewtonQuotientfx,x=c,h=h
c+h−1−c−1h
→limit
Application of Definition 2.2.1
Expression palette: Limit template Type the difference quotient
limh→0fc+h−fch = 12c−1
The difference quotients in Examples 2.2.1-3 each require different algebraic manipulations for the stepwise computation of the limit.
The difference quotient and its simplification
Write the difference quotient and rationalize the numerator.
fc+h−fch
=c+h−1−c−1h
=c+h−1−c−1h⋅c+h−1+c−1c+h−1+c−1
=c+h−1−c−1hc+h−1+c−1
=hhc+h−1+c−1
=1c+h−1+c−1
Since the simplification of the difference quotient has been accomplished (the indeterminate form 0/0 has been eliminated), the limit can be evaluated by simply setting h=0.
Expression palette: Evaluation template Copy/paste simplified difference quotient
1c+h−1+c−1x=a|f(x)h=0 = 12c−1
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