Derivatives of Higher Order dy dx notation If y=f(x),then we may also denote f(x)by d dx Example 1 3x-1 Find d的 for y d 5x+2 Sals,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order dx dy If y = f (x) , then we may also denote by . dy/dx notation Example 1 Find for y = . dy dx 3x – 1 5x + 2 f '(x)
Derivatives of Higher Order Example 2 Find dy fory=(x3+1)(3x5+2x-1) dx Example 2,p.125 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 2, p. 125 Example 2 Find for y = (x 3 + 1) (3 x 5 + 2x – 1). dy dx
Derivatives of Higher Order Example 3 d Find Example 3,p.126 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 3, p. 126 Example 3 Find （t 3 – ）. d dt t t 2 – 1
Derivatives of Higher Order Example 4 du Find for u=x (x+1)(x+2) dt Example 4,p.126 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 4, p. 126 Example 4 Find for u = x (x + 1) (x + 2). du dt
Derivatives of Higher Order Example 5 x2 Find dy/dx at x=0 and x=1 given thaty= 2-4 Example 5,p.126 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 5, p. 126 Example 5 Find dy/dx at x = 0 and x = 1 given that y = . x 2 x 2 – 4
Derivatives of Higher Order Let fx)be a differentiable function.Then the derivative f'x)is also a function. Suppose that f'x)is also differentiable.We can differetiate it and obtain the second derivative d'y d f"(x)=('(x)' or dx Inductively,we can define the n-th derivative by fm(x)=(fm-(x)》 d or Derivatives of higher order,p.127 Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Derivatives of higher order, p. 127 Let f(x) be a differentiable function. Then the derivative f ’(x) is also a function. Suppose that f ’(x) is also differentiable. We can differetiate it and obtain the second derivative or f "(x) = ( f '(x))' = dx dy dx d dx d y 2 2 Inductively, we can define the n-th derivative by or ( ) ( ( ))' ( ) ( 1) f x f x n n− = = − − 1 1 n n n n dx d y dx d dx d y
Derivatives of Higher Order Example 6 Let f(x)=x4-3x1+5.Find f'(x)and f"(x). Example 6,p.127 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 6, p. 127 Example 6 Let f(x) = x 4 – 3x –1 + 5. Find f ’(x) and f ”(x)
Derivatives of Higher Order Example 7 d kx5-4r3+7m)= d (x5-4x3+7x)= dx2 d (x5-4x3+7x)= dx3 Example 7,p.127 Sals,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 7, p. 127 Example 7 (x 5 – 4x 3 + 7x) = (x 5 – 4x 3 + 7x) = (x 5 – 4x 3 + 7x) = d dx d 2 dx2 d 3 dx3
Derivatives of Higher Order Example 8 Finally,we consider y=x-1. Example 8,p.127-128 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 8, p. 127-128 Example 8 Finally, we consider y = x –1
(）-(←1yaxe Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. ( ) 1 1 ( 1 ) ! − − − = − n n nn x n x dxd