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
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Differentiation Rule
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Conditions
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Constant
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Identity
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Power
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is a rational number
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Constant Multiple
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Sum
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Difference
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Product
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Quotient
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Table 2.3.1 Differentiation rules: the operators and prime (#) are used interchangeably, denotes a constant, and and are functions differentiable at
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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
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.