How Do I Find The Derivative Fraction Al Exponents, World Web Math: Fractional Exponents

Suggested Prerequesites: Derivatives of polynomials,Implicit differentiation,The Chain ruleWe know that the Power Rule, an extension of the Product Rule and theQuotient Rule, expressed as

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is valid for any integer exponent n. Whatabout functions with fractional exponents, such asy=x2/3? In this case,y may be expressed as an implicit function of x,y3=x2.Then,

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This is seen to be consistent with the Power Rule forn=2/3.Let”s make a generalization of this example. Anyrational number n can be expressed asp/q for some integers p and nonzero q. Then, fory=xn,

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This is exactly what we would get if we assume the same power ruleholds for fractional exponents as does for integral exponents.

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Notethat we did not need to assume anything about the signs ofp or q, other than the fact that qcannot be zero. Therefore, our power rule can now safely be applied toany rational exponents.The definition of the derivative may also be used, but as the next twoexamples show, the direct use of the definition is often much morecumbersome than the improved Power Rule. Consider the fairly simple case

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From the definition of the derivative,

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in agreement with the Power Rule for n=1/2.For n=–1/2, the definition of the derivative gives

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and a similar algebraic manipulation leads to

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again in agreement with the Power Rule.To see how more complicated cases could be handled, recall theexample above,

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From the definition of the derivative,

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once more in agreement with the Power Rule. This example shouldclearly show that for fractional exponents, using the Power Rule isfar more convenient than resort to the definition of the derivative.

Some examples:

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Exercises:

Find the derivative with respect to x of each of thefollowing functions.

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Solutions to the exercises |Back to the Calculus page |Back to the World wnyrails.org Math top pagewatko

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