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Unformatted text preview: GROUP WORK I, SECTION 3.3
Doing a Lot with a Little . . d . ‘ .
Sect1on 3.3 1ntroduces the Power Rule: E—x" = nx"“‘, where n 15 any real number. The good news IS that
x . this rule, combined with the Constant Multiple and Sum Rules, allows us to take the derivative of even the
most formidable polynomial with ease! To demonstrate this power, try Problem 1: I. A formidable polynomial:
f(x) = x10 + gx9 + %x8 — 5x7 — 0.33x6 + 7rx5 — ﬁx“ — 42
Its derivative: f’(x)= The ability to differentiate polynomials is only one of the things we’ve gained by establishing the Power Rule.
Using some basic deﬁnitions, and a touch of algebra, there are all kinds of functions that can be diﬁerentiated
using the Power Rule. 2. All kinds of functions: 5 _ 1 1 _ x5 — 3,5 + 2
f<x>=€/§+f2 goo—Fax, 1200— ﬂ
Their derivatives: f/ (x) = g’ (x) = h’ (x) = Unfortunately, there are some deceptive functions that look like they should be straightforward applications
of the Power and Constant Multiple Rules, but actually require a little thought. 3. Some deceptive functions: f(x) = (2x)4 gov) = (x3)5 Their derivatives: f/(x)= g’(x)= Doing a Lot with a Little The process you used to take the derivative of the functions in Problem 3 can be generalized. In the ﬁrst case,
f (x) = (2x)4, we had a function that was of the form (kx)”, where k and n were constants (k = 2 and
n = 4). In the second case, g (x) = (x3)5, we had a function of the form (xk)". Now we are going to ﬁnd a
pattern, similar to the Power Rule, that will allow us to ﬁnd the derivatives of these functions as well. 4. Show that your answers to Problem 3 can also be written in this form: f’(x) = 4(2x>3  2 g/(x) = 5 (x3)4 . 3x2 And now it is time to generalize the Power Rule. Consider the two general functions, and try to ﬁnd expres
sions for the derivatives similar in form to those given in Problem 4. You may assume that n is an integer. 5. Two general ﬁmctions: f(x) = (W g(x) = M)” Their derivatives: f I (x)
g” (x) ...
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 Fall '11
 JamesLang
 Calculus, Geometry

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