Calc03_3 - 3.3 Differentiation Rules Colorado National...

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3.3 Differentiation Rules 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.
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We saw that if , . 2 y x = 2 y x = This is part of a pattern. ( 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
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( 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 = ⋅ =
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(Each term is treated separately) ( 29 d du cu c dx dx = constant multiple rule: sum and difference rules: ( 29 d du dv u v dx dx dx + = + ( 29 d du dv u v dx dx dx - = - 4 12 y x x = + 3 4 12 y x = + 4 2 2 2 y x x = - + 3 4 4 dy x x dx = -
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Example: Find the horizontal tangents of: 4 2 2 2 y x x = - + 3 4 4 dy x x dx = - Horizontal tangents occur when slope = zero. 3 4 4 0 x x - = 3 0 x x - = ( 29 2 1 0 x x - = ( 29 ( 29 1 1 0 x x x + - = 0, 1, 1 x = - Plugging the x values into the original equation, we get: 2, 1, 1 y y y = = = (The function is even , so we only get two horizontal tangents.)
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4 2 2 2 y x x = - +
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4 2 2 2 y x x = - + 2 y =
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4 2 2 2 y x x = - + 2 y = 1 y =
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4 2 2 2 y x x = - +
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0, 1, 1 x = - 3 4 4 dy x x dx = -
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Calc03_3 - 3.3 Differentiation Rules Colorado National...

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