BasicDerivativeRules

BasicDerivativeRules - 1 DIFFERENTIATION RULES The...

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1 DIFFERENTIATION RULES The definition of the derivative is nice to prove why certain derivatives are the way they are, but as you have experienced they can be tedious at times. Therefore we state here the short cut differentiation rules that we can use from now on. The proofs of these rules are provided in class and they should be understood. (You would not want to feel guilty using these rules without understanding, would you?) RULE 1: DERIVATIVE OF A CONSTANT If c is a constant, then d c dx = 0. EXAMPLE 1: Find the derivative of f (x) = 6. SOLUTION: f ' (x) = 0 RRULE 2: POWER RULE FOR POSITIVE INTEGERS If n is a positive integer, then d x n dx = n x n -1 . This is the famous “drop-down” rule. EXAMPLE 2: Find the derivative of f (x) = x 4 . SOLUTION: f ' (x) = 4x 4 - 1 = 4x 3 EXAMPLE 3: Find the derivative of f (x) = x. SOLUTION: f (x) = x = x 1 f ' (x) = 1x 1 - 1 = 1x 0 = 1 EXAMPLE 4: Find the derivative of f (x) = x 3 . SOLUTION: f ' (x) = 3x 3 - 1 = 3x 2 RULE 3: CONSTANT MULTIPLE RULE If u is a differentiable function of x, and c is a constant, then d cu dx = c du dx . What this is saying is that we can pull the constant out in front of the function, and then take the derivative of the variable.
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2 EXAMPLE 5: Find the derivative of f (x) = 6x 4 . SOLUTION: f ' (x) = 6(4x 4 - 1 ) = 24x 3 EXAMPLE 6: Find the derivative of f (x) = 7x 9 . SOLUTION: f ' (x) = 7(9x 9 - 1 ) = 63x 8 RULE 4: THE SUM/DIFFERENCE RULE If u and v are differentiable functions of x, then their sum/difference u ± v is differentiable at every point where u and v are both differentiable. As such points,
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This note was uploaded on 04/17/2008 for the course MAC 2311 taught by Professor Noohi during the Spring '08 term at FSU.

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BasicDerivativeRules - 1 DIFFERENTIATION RULES The...

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