second derivative - The Second Derivative OBJECTIVES 1 Find...

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The Second Derivative OBJECTIVES 1) Find the higher derivative of a continuous function. 2) Find relative extrema of a continuous function using the Second-Derivative Test 3) Find the Intervals on which a continuous function concave upward, or concave downward. Higher Derivatives: If a function f has a derivative f , then the derivative of f ,if it exists ,called the second derivative of f , written f ′′ ( x ). The derivative of f ′′ ( x ) , if it exists, is called the third derivative of f , written ( ) f x ′′′ ,..and so on. The second derivative of y = f ( x ) can be written in any of the following notations: 2 '' 2 2 ( ) , , , or [ ( )] x dy f x y D f x dx The third derivative can be written in a similar way. When notation likes () f x gets lengthy, we abbreviate it using a symbol in parentheses. Thus n f x is the n th derivative, for or n 4 . Example 1 Find the second derivative of the function 54 11 3 fx x x x =− + Solution Find the first derivative
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'4 () 5 1 2 1 fx x x =− + 3 Now find the derivative of the first derivative '' ' 3 20 36 () xx d ff dx == 2 Example 2 For 1 x y = , find 2 2 d y dx . Solution Rewrite 1 x y = as 1 yx = 2 dy x dx , and the second derivative 2 3 23 2 2, o r dy x dx x = . Example 3 For , find , and . 21 2 (5 3 ) x + ' y 5 ) y Solution Use the Chain Rule to find ' y 1 ' 12 3 ) ( 2 x x = −+
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To find , ,the second derivative, use the product rule and the chain rule. '' y 21 0 2 12 " 132( 5 3) (2 5) (2 5) ( 5 3 y xx x x =− + + + Factor 12( 5 3) −+ 0 0 2 {11 } 12( 5 2( 5 x x + + + Simplify, 0 2 2 {11( 4 ) } 5 3) 20 25 2 10 6) x x x x ⋅+ + + + 0 2 2 {44 } 12( 5 3) 220 275 2 10 6 x x x x + + = 0 2 {46 } 5 3) 230 281 x x + = DEFINITION: The velocity
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second derivative - The Second Derivative OBJECTIVES 1 Find...

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