Differentiation Formulas THEOREM 3.2.3 DERIVATIVES OF SUMS AND SCALAR MULTIPLES Let a be a real number.If f and g are differentiable at x,then f+g and f are differentiable at x.Moreover. (f+g)(x)=f(x)+g'(x)and (f)'(x)=af'(x). (f-g)'(x)=f'(x)-g(x). Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas
Differentiation Formulas THEOREM 3.2.6 THE PRODUCT RULE Iffandg are differentiable atx,then so is their product,and (f·g)'(x)=f(x)g'(x)+g(x)f'(x). The productrule,p.117 Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas The product rule, p. 117
Differentiation Formulas for each positive integer n (3.2.7) p(x)=x"has derivative p'(x)=nx"-1 Derivatives of powers and polynomials,(3.2.7),(3.2.8),pp.117,118 Main Menu cmew59时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Derivatives of powers and polynomials, (3.2.7), (3.2.8), pp. 117, 118
公 P(x)=anx"+an-1x"-1+…+a2x2+a1x+a0, (3.2.8) then P'(e)=nanx-1+(n-1)an-1x-2+…+2a2x+a1. Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Differentiation Formulas Example 1 Differentiate F(x)=(x3-2x +3)(4x2 +1)and find F(1) Example 1,p.118 Main Menu o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 1, p. 118 Example 1 Differentiate F(x) = (x3 – 2x + 3)(4x2 + 1) and find F’(–1)
Differentiation Formulas Example 2 Differentiate F(x)=(ax+b)(cx +d).where a,b,c,d are constants. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 2 Differentiate F(x) = (ax + b)(cx + d), where a, b, c, d are constants
Differentiation Formulas Example 3 Suppose that g is differentiable at each x and that F(x)=(x3-5x)g(x). Find F(2)given that g(2)=3 and g'(2)=-1. Example 3,p.119 Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 3, p. 119 Example 3 Suppose that g is differentiable at each x and that F(x) = (x 3 – 5x)g(x). Find F’(2) given that g(2) = 3 and g’(2) = –1
Differentiation Formulas THEOREM 3.2.9 THE RECIPROCAL RULE Ifg is differentiable at x and g(x)0,then 1/g is differentiable at x and g'(x) (x)= [g(x)2 The reciprocal rule,p.119-120 Main Menu
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas The reciprocal rule, p. 119-120
Example 4 Differentiatef(x)= and tind 56 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. 5 x 2 6 x 1 2 Example 4 Differentiate f (x) = – and find f ’( )
Differentiation Formulas Example 5 1 Differentiate fx)=- where a,b,c are constants. ax2+bx+c Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 5 Differentiate f(x) = , where a, b, c are constants. 1 ax2 + bx + c