# lecture06 - Lecture 6 Rules of Differentiation Velocity and...

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Unformatted text preview: Lecture 6 - Rules of Differentiation, Velocity and Marginals 6.1 Rules for Differentiation We would like to avoid the limit calculation for the derivative entirely, so we state some rules for finding the derivative directly. 1. Constant Rule: d d x ( c ) = 0 for any constant c 2. Power Rule: If n is any real number, d d x ( x n ) = nx n- 1 So, for example, d d x ( x ) = 1 d d x ( x 7 ) = 7 x 6 3. Constant Multiple Rule: d d x [ cf ( x )] = c d d x f ( x ) 4. Sum Rule: d d x ( f ( x ) + g ( x )) = d d x f ( x ) + d d x g ( x ) Examples: 1. Find the derivative of f ( t ) = t 2 / 3 + 3 t- 3. solution: We use a combination of the sum rule, constant multiple rule, and power rule: d d t ( t 2 / 3 + 3 t- 3) = 2 3 t 2 / 3- 1 + 3 t 1- 1 + 0 = 2 3 t- 1 / 3 + 3 2. Find the derivative of f ( x ) = 4 x 3 . solution: Here we use the power rule and contant multiple rule: d d x 4 x 3 = d d x 4 x- 3 = 4 d d x x- 3 = 4(- 3) x- 3- 1 =- 12 x- 4 =- 12 x 4 6.4 Product and Quotient Rules In addition to the rules above, we have rules for taking the derivatives of the products and quotients of functions. Product Rule d d x [ f ( x ) g ( x )] = f ( x ) g ( x ) + f ( x ) g ( x ) Quotient Rule d d x f ( x ) g ( x ) = f ( x ) g ( x )- g ( x ) f ( x ) [ g ( x )] 2 Examples 1. Differentiate f ( x ) = ( x 2 + 1)(2 x + 5) (differentiate means “find the derivative of”). solution: There are two ways to do this. The first is to multiply the expression out and then take the derivative term-by-term. Since we have just given the product rule, let’s do it that way instead. f ( x ) = ( x 2 + 1) (2 x + 5) + ( x 2 + 1)(2 x + 5) = (2 x )(2 x + 5) + ( x 2 + 1)(2) = 6 x 2 + 10 x + 2 2. Find the derivative of g ( x ) = ( 1 x + 3 ) ( 1 x 2- 4 ) . solution: Once again we could do it two different ways, but let’s use the product rule. g ( x ) = 1 x + 3 1 x 2- 4 + 1 x + 3 1 x 2- 4 =- 1 x 2 1 x 2- 4 + 1 x + 3- 2 x 3 =- 1 x 4 + 4 x 2- 2 x 4- 6 x 3 =- 3 x 4- 6 x 3 + 4 x 2 3. Find the derivative of h ( x ) = √ x x +1 ....
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lecture06 - Lecture 6 Rules of Differentiation Velocity and...

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