Chapter 2: Differentiation
Section 2.2: Precise Definition of the Derivative
Example 2.2.1
Apply Definition 2.2.1 to fx=x3, to obtain f′c, the derivative at the point c,fc.
Solution
Define the function f
Control-drag fx=… Context Panel: Assign Function
fx=x3→assign as functionf
Method 1
Write the mathematical notation for the derivative
Context Panel: Evaluate and Display Inline
f′c = 3c2
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
x3
→differentiate w.r.t. x
3x2
→evaluate at point
3c2
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: Evaluate at a Point≻h=0 (The difference quotient is returned simplified, so the limit is found by setting h to zero.)
NewtonQuotientfx,x=c,h=h
3c2+3ch+h2
Application of Definition 2.2.1
Expression palette: Limit template Type the difference quotient
limh→0fc+h−fch = 3c2
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 numerator of the difference quotient; Press the Enter key.
Context Panel: Expand≻Expand
Context Panel: Factor
Referencing via equation label (Control + L), form the difference quotient. Press the Enter key. Context Menu: Label≻Label Reference
Context Panel: Evaluate at a Point≻h=0
fc+h−fc
c+h3−c3
= expand
3c2h+3ch2+h3
= factor
h3c2+3ch+h2
h
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