mate 3031 - recinto universitario de...

MATE 3031

Dr. Pedro Vásquez

UPRM

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MATE 3031

The Product and Quotient RulesThe Product RuleBy analogy with the Sum and Di§erence Rules, one might be tempted toguess, that the derivative of a product is the product of the derivatives.We can see, however, that this guess is wrong by looking at a particularexample.Let f (x) = x2 and g(x) = x3

Then the Power Rule givesf 0(x) = and g 0(x) =But (fg)(x) = x5, so (fg)0(x) = .Thus: (fg)0 f 0g 0

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The correct formula was discovered by Leibniz and is called the ProductRule.Before stating the Product Rule, let’s see how we might discover it.We start by assuming that u = f (x) and v = g(x) are both positivedi§erentiable functions. Then we can interpret the product uv as an areaof a rectangle (see Figure).

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MATE 3031

.If x changes by an amount Dx , then the corresponding changes in u and vare

Du = f (x + Dx)− f (x) Dv = g(x + Dx)− g(x)and the new value of the product, (u + Du)(v + Dv), can be interpretedas the area of the large rectangle in Figure (provided that Du and Dvhappen to be positive).The change in the area of the rectangle is

D(uv) = (u + Du)(v + Dv)− uv = uDv + vDu + DuDv (1)= the sum of the three shaded areas

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If we divide by Dx , we get

If we now let Dx ! 0, we get the derivative of uv :

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Ejmplos.Halle la derivada de:1. f (x) = x2ex

2. f (x) = (1− ex ) (x + ex )

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3. f (x) =!x3 − x

"ex

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The Quotient RuleWe find a rule for di§erentiating the quotient of two di§erentiablefunctions u = f (x) and v = g(x) in much the same way that we foundthe Product Rule.If x , u, and v change by amounts Dx , Du, and Dv , then the correspondingchange in the quotient u/v is

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so

As Dx ! 0, Dv ! 0 also, because v = g(x) is di§erentiable and thereforecontinuous.Thus, using the Limit Laws, we get

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MATE 3031

In words, the Quotient Rule says that the derivative of a quotient is thedenominator times the derivative of the numerator minus the numeratortimes the derivative of the denominator, all divided by the square of thedenominator.

4. f (x) =ex

1− ex

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5.f (x) =ax

bx2 + cx3

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6.f (x) =4+ xxex

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7. Halle f 0 y f 00, si:a. f (x) =

pxex

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b. f (x) = xx 2−1

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36. The curve y = x/!1+ x2

"is called a serpentine, find an equation of

the tangent line to this curve at the point (3, 0.3)

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44.

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46.

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50.

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Table of Di§erentiation Formulas

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