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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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