121616949-math.59 - f x = x n Here n is a number of any...

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3 Rules For Finding Derivatives It is tedious to compute a limit every time we need to know the derivative of a function. Fortunately, we can develop a small collection of examples and rules that allow us to compute the derivative of almost any function we are likely to encounter. Many functions involve quantities raised to a constant power, such as polynomials and more complicated combinations like y = (sin x ) 4 . So we start by examining powers of a single variable; this gives us a building block for more complicated examples. We start with the derivative of a power function,
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Unformatted text preview: f ( x ) = x n . Here n is a number of any kind: integer, rational, positive, negative, even irrational, as in x π . We have already computed some simple examples, so the formula should not be a complete surprise: d dx x n = nx n-1 . It is not easy to show this is true for any n . We will do some of the easier cases now, and discuss the rest later. The easiest, and most common, is the case that n is a positive integer. To compute the derivative we need to compute the following limit: d dx x n = lim Δ x → ( x + Δ x ) n-x n Δ x . 45...
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