Chapter 2: Differentiation
Section 2.3: Differentiation Rules
Introduction
Evaluation of derivatives using the definition hinges on the ability to evaluate the limit of the difference quotient. This is tedious and time consuming. Even worse, the limits can be extremely difficult to evaluate. The differentiation rules are a collection of general rules for computing derivatives formed from polynomials, powers, roots, and constant multiples, sums, differences, products, quotients of these types of functions. Compositions of functions are handled separately in the next section, on the Chain rule.
Be careful not to confuse the limit laws and differentiation rules. While many are analogous, others are completely different.
Like the tutor, the tutor is an excellent tool for learning the names and function of the differentiation rules and mastering how they can be applied to evaluate a derivative. The three examples provided in this section illustrate the use of this tool.
Essentials
The Differentiation Rules
In general, derivatives can be evaluated by applying the definition, as in Examples 2.2.1 - 3 in the previous section. The first few rules listed in Table 2.3.1 are fairly simple, almost obvious. But, starting with the Product rule, the rules are not so obvious and are quite different from the corresponding limit laws.
Rule Name
Differentiation Rule
Conditions
Constant
ddxk= 0
Identity
dxdx=1
Power
ddxxn= n xn−1
n≠0 is a rational number
Constant Multiple
kfx′= k f′x
Sum
fx+gx′= f′x+g′x
Difference
fx−gx′= f′x− g′x
Product
fxgx′= fxg′x+gxf′x
Quotient
fxgx′=gxf′x−fxg′xg2x
gx≠0
Table 2.3.1 Differentiation rules: the operators ddx and prime (#) are used interchangeably, k denotes a constant, and f and g are functions differentiable at x
Note the conditions listed in the third column. If the conditions for a rule are not satisfied, the rule cannot be used to evaluate a derivative.
The Constant Multiple rule simply says that when differentiating the product of a function times a constant multiple, the derivative of the function is computed and multiplied by the constant.
The Sum and Difference rules simply say that the derivative of a sum or difference is the sum or difference of the derivatives.
The Product rule may best be learned as "the first times the derivative of the second plus the second times the derivative of the first."
In actuality, the pattern being followed is best seen if the derivative of a product of three factors is written. The derivative of a product of three factors is given by
fgh′= f′ g h + f g′ h + f g h′
The product of three factors is written three times, in each of these three terms just one of the functions is differentiated, and the terms are added. With this pattern in mind, the Product rule for two factors is seen to obey exactly this schematic, but the vocalization in terms of "first" and "second" makes articulating and applying the rule much simpler.
Finally, the Quotient rule can be stated as "denominator times derivative of the numerator, minus numerator times derivative of the denominator, all over the denominator squared." (With this version of the rule, the letters DN lead off the vocalization. Remembering that D stands for denominator and that DNA is the biological key to life, no student should ever reverse the roles of numerator and denominator in the Quotient rule.)
In some classrooms, students learn the Quotient rule with "top" and "bottom" replacing "numerator" and "denominator," respectively. In either event, articulating a pattern in language is far easier than trying to memorize it as a string of symbols.
Proofs of the Differentiation Rules
Constant Rule
Let fx=k, where k is a constant. Apply Definition 2.2.1.
f′x
=limh→0fx+h−fxh
=limh→0k−kh
=limh→00h
=limh→00
=0
This result is eminently reasonable because the graph of a constant function is a horizontal line, and the slope of such a line is zero.
Identity Rule
Let fx=x and apply Definition 2.2.1.
=limh→0x+h−xh
=limh→0hh
=limh→01
=1
This result is eminently reasonable because the graph of fx=x is a straight line with slope 1.
Power Rule
Apply the Binomial theorem to x+hn, where n is a positive integer.
x+hn
=∑k=0nnkxn hn−k
=xn+n h xn−1+n2 h2 xn−2+n3 h3 xn−3+⋯
=xn+n h xn−1+h2n2xn−2+n3h xn−3+⋯
=xn+n h xn−1+Oh2
The Landau order symbol Ohp denotes a quantity that is a multiple of h2. In particular, if g=Oh2, then g/h=Oh, and g/h=Oh→0 as h→0.
Let fx=xn, where n is a positive integer. Apply Definition 2.2.1.
=limh→0x+hn−xnh
=limh→0xn+n h xn−1+Oh2−xnh
=limh→0n h xn−1+Oh2h
=n xn−1+limh→0Oh
=n xn−1+0
=n xn−1
To extend the Power rule to integers n<0, apply the Quotient rule to xn=1/x−n, remembering that −n is a positive integer.
ddxxn
=ddx1x−n
=x−nddx1 −1⋅ddxx−nx−n2
=0−−nx−n−1x−2 n
=nx−n−1+2 n
To extend the Power rule to rational numbers a=p/q, write y=xa as yq=xp and differentiate both sides, using the Chain rule (see Section 2.4) on the left, remembering that y=yx.
Differentiate via the Chain rule
ddxyq
=ddxxp
q yq−1y′x
=p xp−1
Solve for y′x
y′x
=p xp−1q yq−1
=p xp−1q xp/qq−1
=p xp−1q xp−p/q
=a xp−1−p−p/q
=a xp/q−1
=a xa−1
Note: The typical calculus text will state the Power rule and declare it applicable for all real exponents, but postpone the full proof until additional differentiation techniques have first been established. Proofs based on implicit differentiation (Section 2.5) or logarithmic differentiation (Table 2.6.2) amount to a proof based on the Chain rule; hence, the choice made above.
Constant Multiple Rule
Let Fx=k fx and apply Definition 2.2.1 to F.
F′x
=limh→0Fx+h−Fxh
=limh→0k fx+h−k fxh
=k limh→0fx+h−fxh
=k f′x
Sum Rule
The Sum rule for derivatives follows from the Sum rule for limits.
ddxfx+gx
=limh→0fx+h+gx+h−fx+gxh
=limh→0fx+h−fx+gx+h−gxh
=limh→0fx+h−fxh+limh→0gx+h−gxh
=f′x+g′x
Difference Rule
The Difference rule for derivatives follows from the Difference rule for limits.
ddxfx−gx
=limh→0fx+h−gx+h−fx−gxh
=limh→0fx+h−fx−gx+h−gxh
=limh→0fx+h−fxh−limh→0gx+h−gxh
=f′x−g′x
Product Rule
The derivative of a product is not the product of the derivatives!
ddxfxgx
=limh→0fx+hgx+h−fxgxh
=limh→0fx+hgx+h−fx+hgx+fx+hgx−fxgxh
=limh→0fx+hgx+h−gx+fx+h−fxgxh
=limh→0fx+hgx+h−gxh+gxlimh→0fx+h−fxh
=limh→0fx+h limh→0gx+h−gxh+gxlimh→0fx+h−fxh
=fx g′x+gx f′x
The device of adding and subtracting the same term is used often in analysis. Here, the term added and subtracted to the numerator in the second line is fx+hgx.
To go from the fourth line to the fifth, the Product rule for limits is applied. Since f is differentiable, it is continuous, so its limit exists, and fx+h→fx as h→0.
Quotient Rule
The derivative of a quotient is not the quotient of the derivatives!
=limh→0fx+hgx−fxgx+hh gxgx+h
=limh→0fx+hgx−fxgx+fxgx−fxgx+hhgxgx+h
=limh→0fx+h−fxgx−fxgx+h−gxh gxgx+h
=limh→0fx+h−fxgxhgxgx+h−limh→0fxgx+h− gxhgxgx+h
=limh→0fx+h−fxhgx+h−fxgxlimh→0gx+h−gxh gx+h
=limh→0fx+h−fxhlimh→0gx+h−fxgxlimh→0gx+h−gxhlimh→0gx+h
=f′xgx−fxgx g′xgx
=gxf′x−fxg′xg2x
In the third line, the term fxgx is added and subtracted. In going from the sixth line to the seventh, the Quotient rule for limits is applied. The very last line is the result of adding fractions over a common denominator.
Précis
Table 2.3.2 summarizes the key points in Section 2.3.
The Differentiation rules in Table 2.3.1 provide the tools necessary for differentiating a function without explicitly evaluating a limit. With these rules it is possible to differentiate any polynomial or rational function.
Note the Product and Quotient rules since they do not parallel the corresponding Limit laws. The derivative of a product is not the product of the derivatives. The derivative of a quotient is not the quotient of the derivatives.
The Power rule applies to any integer.
Table 2.3.2 Key points in Section 2.3
Examples
Example 2.3.1
Apply the rules in Table 2.3.1 to obtain the derivative of fx=x3−3x2+x+3.
Example 2.3.2
Apply the rules in Table 2.3.1 to obtain the derivative of ft=t−1t2+2.
Example 2.3.3
Apply the rules in Table 2.3.1 to obtain the derivative of fx=5 x2−7 x+12 x3−3x2+x+3.
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