Chapter 2: Differentiation
Section 2.2: Precise Definition of the Derivative
Apply Definition 2.2.1 to fx=x−1, to obtain f′c, c>1.
Define the function f
Context Panel: Assign Function
fx=x−1→assign as functionf
Write the mathematical notation for the derivative
Context Panel: Evaluate and Display Inline
f′c = 12⁢c−1
Type fx and press the Enter key.
Context Panel: Differentiate≻With Respect To≻x
Context Panel: Evaluate at a Point≻x=c
→differentiate w.r.t. x
→evaluate at point
Expression palette: Differentiation template
Apply to fx and press the Enter key.
Evaluate at x=c as above.
ⅆⅆ x fx
Tools≻Load Package: Student Calculus 1
Apply the NewtonQuotient command.
Context Panel: Limit
Select h as the variable
Application of Definition 2.2.1
Expression palette: Limit template
Type the difference quotient
limh→0fc+h−fch = 12⁢c−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.
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 = 12⁢c−1
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