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3_1 - Section 3.1 Derivatives of Polynomials and...

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Section 3.1: Derivatives of Polynomials and Exponential Functions Instructor: Ms. Hoa Nguyen ([email protected]) Power Rule If f ( x ) = x n ( n is any real number) then f ( x ) = nx n - 1 . Example : f ( x ) = x 1000 f ( x ) = f ( t ) = t 3 f ( t ) = f ( x ) = x f ( x ) = f ( x ) = c (where c is a constant) f ( x ) = f ( r ) = - 1 r 2 f ( r ) = f ( x ) = x f ( x ) = f ( x ) = 3 x 2 f ( x ) = f ( x ) = x x f ( x ) = Remark : f = df dx (Leibniz notation) Compute Derivatives of “New” Functions “New” functions are formed from “old” functions by addition, subtraction, multiplication or division. If we know the derivatives of “old” functions, we can compute the derivatives of the “new” functions based on the following rules: Assume that the derivatives of f, g exist (i.e., f, g are differentiable), then The constant multiple rule : d dx [ cf ] = c df dx ( cf ) = cf ( c is a constant) The sum or difference rule : d dx [ f ± g ] = df dx ± dg dx ( f ± g ) = f ± g We will learn about product or quotient rule on the next section. Example 5 of Section 3.1 (textbook): Example 6 of Section 3.1 (textbook): Example 7 of Section 3.1 (textbook): Exponential function If f ( x ) = a x ( a > 0 , a = 1
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