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Unformatted text preview: 2-15SECTION 2.2..The Derivative1592.2THE DERIVATIVEIn section 2.1, we investigated two seemingly unrelated concepts: slopes of tangent linesand velocity, both of which are expressed in terms of thesamelimit. This is an indicationof the power of mathematics, that otherwise unrelated notions are described by thesamemathematical expression. This particular limit turns out to be so useful that we give it aspecial name.DEFINITION 2.1Thederivativeof the functionf(x) atx=ais defined asf(a)=limh→f(a+h)−f(a)h,(2.1)provided the limit exists. If the limit exists, we say thatfisdifferentiableatx=a.An alternative form of (2.1) isf(a)=limb→af(b)−f(a)b−a.(2.2)(See exercise 49 in section 2.1.)EXAMPLE 2.1Finding the Derivative at a PointCompute the derivative off(x)=3x3+2x−1 atx=1.SolutionFrom (2.1), we havef(1)=limh→f(1+h)−f(1)h=limh→3(1+h)3+2(1+h)−1−(3+2−1)h=limh→3(1+3h+3h2+h3)+(2+2h)−1−4hMultiply out and cancel.=limh→11h+9h2+3h3hFactor out commonhand cancel.=limh→(11+9h+3h2)=11.Suppose that in example 2.1 we had also needed to findf(2) andf(3).Must we nowrepeat the same long limit calculation to find each off(2) andf(3)? Instead, we computethe derivative without specifying a value forx, leaving us with a function from which wecan calculatef(a) for anya, simply by substitutingaforx.EXAMPLE 2.2Finding the Derivative at an Unspecified PointFind the derivative off(x)=3x3+2x−1 at an unspecified value ofx. Then, evaluatethe derivative atx=1,x=2 andx=3.160CHAPTER 2..Differentiation2-16SolutionReplacingawithxin the definition of the derivative (2.1), we havef(x)=limh→f(x+h)−f(x)h=limh→3(x+h)3+2(x+h)−1−(3x3+2x−1)h=limh→3(x3+3x2h+3xh2+h3)+(2x+2h)−1−3x3−2x+1hMultiply outand cancel.=limh→9x2h+9xh2+3h3+2hhFactor outcommonhand cancel.=limh→(9x2+9xh+3h2+2)=9x2+++2=9x2+2.Notice that in this case, we have derived a newfunction, f(x)=9x2+2.Simplysubstituting in forx, we getf(1)=9+2=11 (the same as we got in example 2.1!),f(2)=9(4)+2=38 andf(3)=9(9)+2=83.Example 2.2 leads us to the following definition.DEFINITION 2.2Thederivativeoff(x) is the functionf(x) given byf(x)=limh→f(x+h)−f(x)h,(2.3)provided the limit exists. The process of computing a derivative is calleddifferentiation.Further,fis differentiable on an intervalIif it is differentiable at every point inI.In examples 2.3 and 2.4, observe that the name of the game is to write down the defininglimit and then to find some way of evaluating that limit (which initially has the indeterminateform).EXAMPLE 2.3Finding the Derivative of a Simple Rational FunctionIff(x)=1x(x=0),findf(x).SolutionWe havef(x)=limh→f(x+h)−f(x)h=limh→1x+h−1xhSincef(x+h)=1x+h.=limh→x−(x+h)x(x+h)hAdd fractions and cancel....
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- Derivative, lim