Differentiating the Trigonometric Functions The derivative of the sine function is the cosine function: (3.6.1) d床(sin)=cosx. Basic formulas,(3.6.1),(3.6.2),Example 1,p.142 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Basic formulas, (3.6.1), (3.6.2), Example 1, p. 142
The derivative of the cosine function is the negative of the sine function: (3.6.2) (cosx)=-sinx. dx Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Example 1 Differentiate f(x)=cosx sinx. Main Meny c8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example 1 Differentiate f ( x ) = cos x sin x
Differentiating the Trigonometric Functions The derivative of the tangent function is the secant squared: d (3.6.3) (tanx)=see2x. The derivatives of the other trigonometric functions are as follows: d (cotx)=-csc2x, dx (3.6.4) -(secx)=secx tanx, dx d -(cscx)=-cscx cotx. dx Basic formulas(3.6.3),(3.6.4),p.143 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Basic formulas (3.6.3), (3.6.4), p.143
Differentiating the Trigonometric Functions Example 2 Find f'(z/4)for f(x)=x cot x. Example 2,p.143 Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 2, p.143 Example 2 Find f ’(π/4) for f(x) = x cot x
Differentiating the Trigonometric Functions Example 3 -sec x dx L tanx Example 3,p.143 Main Menu o墙8
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 3, p.143 Example 3 Find [ ]. d dx 1 – sec x tan x
Differentiating the Trigonometric Functions Example 4 Find an equation for the line tangent to the curve y=cosx at the point where x=π/3. Example 4,p.143 Main Menu cmew59时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 4, p.143 Example 4 Find an equation for the line tangent to the curve y = cos x at the point where x = π/3
Differentiating the Trigonometric Functions Example 5 Set f(x)=x+2 sin x.Find the numbers x in the open interval (0,2z) at which (a)f'(x)=0,(b)f'(x)>0,(c)f'(x)<0. Example 5,p.144 Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 5, p.144 Example 5 Set f ( x ) = x + 2 sin x. Find the numbers x in the open interval (0, 2π) at which (a) f ’(x) = 0, (b) f ’(x) ＞ 0, (c) f ’(x) ＜ 0
Differentiating the Trigonometric Functions The Chain Rule Applied to the Trigonometric Functions If f is a differentiable function ofu and u is a differentiable function ofx,then,as you saw in Section 3.5, 品rel=品an- 、du =f'(u dx dx Written in this form,the derivatives ofthe six trigonometric functions appear as follows: d du d du sin0）=cos"a cos0）=-sin】 d du du (3.6.5) (tanu）=sec2 dx dr (cotu)=-csc2u dx dx d du d du -(sec u)=sec u tan u -(csc u)=-csc u cotu dx dx dx The chain rule and the trig functions,(3.6.5),p.144 Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions The chain rule and the trig functions, (3.6.5), p. 144
Differentiating the Trigonometric Functions Example 6 d -(cos 2x) dx Example 6,p.144 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 6, p. 144 Example 6 d dx (cos 2x)