121616949-math.61

# 121616949-math.61 - 3.1 The Power Rule 47 least the power 2...

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3.1 The Power Rule 47 least the power 2. Now let’s look at the limit: d dx x n = lim Δ x 0 ( x + Δ x ) n - x n Δ x = lim Δ x 0 x n + nx n - 1 Δ x + a 2 x n - 2 Δ x 2 + · · · + a n - 1 x Δ x n - 1 + Δ x n - x n Δ x = lim Δ x 0 nx n - 1 Δ x + a 2 x n - 2 Δ x 2 + · · · + a n - 1 x Δ x n - 1 + Δ x n Δ x = lim Δ x 0 nx n - 1 + a 2 x n - 2 Δ x + · · · + a n - 1 x Δ x n - 2 + Δ x n - 1 = nx n - 1 . Now without much trouble we can verify the formula for negative integers. First let’s look at an example: EXAMPLE 3.2 Find the derivative of y = x - 3 . Using the formula, y = - 3 x - 3 - 1 = - 3 x - 4 . Here is the general computation. Suppose n is a negative integer; the algebra is easier to follow if we use n = - m in the computation, where m is a positive integer. d dx x n = d dx x - m = lim Δ x 0 ( x + Δ x ) - m - x - m Δ x = lim Δ x 0 1 ( x x ) m - 1 x m Δ x = lim Δ x 0 x m - ( x + Δ x ) m ( x + Δ x ) m x m Δ x = lim Δ x 0 x m - ( x m + mx m - 1 Δ x + a 2 x m - 2 Δ x 2 + · · · + a m - 1 x Δ x m - 1 + Δ x m ) ( x + Δ x ) m x m Δ x = lim Δ x 0 - mx m - 1 - a 2 x m - 2 Δ x - · · · - a m -
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Unformatted text preview: m x m =-mx m-1 x m x m =-mx m-1 x 2 m =-mx m-1-2 m = nx-m-1 = nx n-1 . We will later see why the other cases of the power rule work, but from now on we will use the power rule whenever n is any real number. Let’s note here a simple case in which the power rule applies, or almost applies, but is not really needed. Suppose that f ( x ) = 1; remember that this “1” is a function, not “merely” a number, and that f ( x ) = 1 has a graph that is a horizontal line, with slope zero everywhere. So we know that f ± ( x ) = 0....
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