Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Calculus 12 – Ch. 2 Derivatives Lesson 8: Higher Derivatives For any function y = f (x) , y! , (simply put: SLOPE) dy or f !(x) indicates the rate of change of y with respect to x dx We can also take the derivative of the derivative d2y Notation: y!!, f !!(x) or dx 2 And taking the derivative again: y!!!, f !!!(x) or d 3y dx 3 Etc. The second derivative is the rate of change of the slope of the tangent line with respect to x The second derivative of a position-time function represents acceleration (we will see more of in Ch. 3) Examples 1) Find y! and y!! for each of the following: a) y = 4x + 3x ! 8 2 3 3 b) y = x ! 2 + x 2 x 1 y! = 8x + 3 1 1 1 !2 3 x + 6x !3 + x 2 2 2 1 6 3 = + 3+ x 2 x x 2 y" = y!! = 8 3 y = x 2 ! 3x !2 + x 2 3 1 1 " 3 " y!! = " x 2 "18x "4 + x 2 4 4 1 18 3 =" 3 " 4 + x 4 x 4x 2 c) y = ( 2 ! x 2 ) 10 d) y = y = ( 3x + 4 ) y! = 10 ( 2 " x 2 ) ("2x ) 9 = "20x ( 2 " x 2 ) y" = ! 9 =! ( ) + "20x ( 9 )( 2 " x ) ( "2x ) = "20 ( 2 " x ) #( 2 " x ) " 18x % $ & = "20 ( 2 " x ) ( 2 " 19x ) y!! = "20 2 " x 2 1 3x + 4 9 2 8 2 8 2 1 2 8 2 y!! = 2 = ! 1 2 3 1 ! ( 3x + 4 ) 2 (3) 2 3 3 2 ( 3x + 4 ) 2 5 9 " ( 3x + 4 ) 2 ( 3) 4 27 5 4 ( 3x + 4 ) 2 2) Use implicit differentiation to find y! and y!! for each of the following: a) y 2 = x 2 + 2x b) y 2 + 2y = 2x +1 2yy! + 2 y! = 2 2yy! = 2x + 2 y! = y! ( y +1) = 1 x +1 y y! = 1 y +1 y! = ( x + 1) y "1 y!! = y "1 + ( x + 1) ( "1) y "2 y! = y + ( x + 1) ( "1) y "1 = 1 ( x + 1) " y y3 "2 ( x + 1) y 2 y 2 " ( x + 1) = y3 y! = ( y +1) "1 y!! = "1( y +1) y! "2 = "1( y +1) 2 x 2 + 2x " (x 2 + 2x + 1) y3 1 =" 3 y = "1 = "1 ( y +1) 3 "2 ( y +1) "1 c) x 3 + y 3 = 5 3x 2 + 3y 2 y! = 0 3y 2 y! = "3x 2 y! = " x2 y2 y! = "x 2 y"2 y!! = "2xy"2 " x 2 ("2 ) y"3 y! # x2 & = "2xy"2 + 2x 2 y"3 %" 2 ( $ y ' =" = = = 2x 2x 4 " 5 y2 y "2xy 3 " 2x 4 y5 "2x ( y 3 + x 3 ) y5 "10x y5