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# Calc03_3 - 3.3 Rules for Differentiation Colorado National...

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3.3 Rules for Differentiation Colorado National Monument Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003

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If the derivative of a function is its slope, then for a constant function, the derivative must be zero. ( 29 0 d c dx = example: 3 y = 0 y = The derivative of a constant is zero.
If we find derivatives with the difference quotient: ( 29 2 2 2 0 lim h x h x d x dx h + - = ( 29 2 2 2 0 lim h x xh h x h + + - = 2 x = ( 29 3 3 3 0 lim h x h x d x dx h + - = ( 29 3 2 3 3 0 3 lim h x x h xh h x h + + + - = 2 3 x = 1 1 1 1 2 1 1 5 10 10 5 1 (Pascal’s Triangle) 2 4 d x dx ( 29 4 2 2 3 4 4 0 6 4 lim h x x h x h xh h x h + + + + - = 3 4 x = 2 3 We observe a pattern: 2 x 2 3 x 3 4 x 4 5 x 5 6 x

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( 29 1 n n d x nx dx - = examples: ( 29 4 f x x = ( 29 3 4 f x x = 8 y x = 7 8 y x = power rule We observe a pattern: 2 x 2 3 x 3 4 x 4 5 x 5 6 x
( 29 d du cu c dx dx = examples: 1 n n d cx cnx dx - = constant multiple rule: 5 4 4 7 7 5 35 d x x x dx = = When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

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