Chapter 2: Differentiation
Section 2.4: The Chain Rule
Obtain the second derivative of Fx=x4+15.
Tools≻Load Package: Student Calculus 1
Context Panel: Assign Function
Fx=x4+15→assign as functionF
F″x = 320⁢x4+13⁢x6+60⁢x4+14⁢x2= factor 20⁢x2⁢19⁢x4+3⁢x4+13
Annotated stepwise calculation of the second derivative
Expression palette: Differentiation template
Apply to F′x.
Context Panel: Student Calculus1≻All Solution Steps (See Figure 2.4.4(a), below.)
ⅆⅆ x F′x→show solution stepsDifferentiation Stepsⅆⅆx20⁢x4+14⁢x3▫1. Apply the constant multiple rule to the term ⅆⅆx20⁢x4+14⁢x3◦Recall the definition of the constant multiple ruleⅆⅆx⁢f⁡x=⁢ⅆⅆxf⁡x◦This means:ⅆⅆx20⁢x4+14⁢x3=We can rewrite the derivative as:20⁢ⅆⅆxx4+14⁢x3▫2. Apply the product rule◦Recall the definition of the product ruleⅆⅆxf⁡x⁢g⁡x=ⅆⅆxf⁡x⁢g⁡x+f⁡x⁢ⅆⅆxg⁡xf⁡x=x4+14g⁡x=x3This gives:20⁢ⅆⅆxx4+14⁢x3+20⁢x4+14⁢ⅆⅆxx3▫3. 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:20⁢ⅆⅆxx4+14⁢x3+60⁢x4+14⁢x2▫4. Apply the chain rule to the term x4+14◦Recall the definition of the chain ruleⅆⅆxf⁡g⁡x=f'⁡g⁡x⁢ⅆⅆxg⁡x◦Outside functionf⁡v=v4◦Inside functiong⁡x=x4+1◦Derivative of outside functionⅆⅆvf⁡v=4⁢v3◦Apply compositionf'⁡g⁡x=4⁢x4+13◦Derivative of inside functionⅆⅆxg⁡x=4⁢x3◦Put it all togetherⅆⅆxf⁡g⁡x⁢ⅆⅆxg⁡x=This gives:320⁢x4+13⁢x6+60⁢x4+14⁢x2
The first derivative is a product. The second derivative is therefore obtained with the Product rule. But one of the factors in this product is a composition, so when the derivative of that factor is calculated, the Chain rule must be used.
The rules for differentiation can be accessed via the Context Panel, as per Figure 2.4.4(a).
Figure 2.4.4(a) Context Panel access to Differentiation Rules
The following is a summary of the computation of the second derivative, given that the first derivative is known.
=ddx20 x3 x4+14
=20 ddxx3 x4+14
=20x34⋅x4+13⋅4 x3+x4+14⋅3 x2
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