lecture13 - Lecture 13: Basic Rules of Dierentiation...

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Lecture 13: Basic Rules of Difierentiation Polynomials and Exponentials (Chapter 3, Section 13) Derivative of a Constant If c is a constant, then d dx ( c ) = 6 - ? ± Power functions of the form f ( x ) = x n 1) d dx ( x ) = 6 - ? ±
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2 2) If n is a positive integer, d dx ( x n ) = To prove this, we need the formula x n ¡ a n = ( x ¡ a )( x n ¡ 1 + x n ¡ 2 a + x n ¡ 3 a 2 + ::: + xa n ¡ 2 + a n ¡ 1 ) NOTE: We can also prove this from the deflnition f 0 ( x ) = lim h ! 0 f ( x + h ) ¡ f ( x ) h and the Binomial Theorem.
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3 This result extends to all real numbers. Power Rule For any real number r , d dx ( x r ) = rx r ¡ 1 . ex. Find the following derivatives: 1) d dx ± 2 = 2) d dx ( x 125 ) = 3) d dx ± 1 x 5 =
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4 ex. Find f 0 ( x ) if f ( x ) = x 4 p x 3 . The next rules, based on the limit laws, allow us to flnd derivatives of some combinations of functions. Constant Multiple Rule If c is a constant and f is difierentiable then d dx ( cf ( x )) =
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5 Sum and Difierence Rules If f and g are both difierentiable, d dx [ f ( x ) § g ( x )] = This can be extended to We can now flnd the derivative of any polynomial function.
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lecture13 - Lecture 13: Basic Rules of Dierentiation...

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