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File: Differentiation Pdf 169644 | Derivative Of Trigonometric Functions
4100 awl thomas ch03p147 243 8 19 04 11 16 am page 183 3 4 derivatives of trigonometric functions 183 3 4 derivatives of trigonometric functions many of the phenomena ...

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           4100 AWL/Thomas_ch03p147-243  8/19/04  11:16 AM  Page 183
                                                                                                            3.4   Derivatives of Trigonometric Functions   183
                             3.4               Derivatives of Trigonometric Functions
                                                                   Many of the phenomena we want information about are approximately periodic (electro-
                                                                   magnetic fields, heart rhythms, tides, weather). The derivatives of sines and cosines play a
                                                                   key role in describing periodic changes. This section shows how to differentiate the six ba-
                                                                   sic trigonometric functions.
                                                                   Derivative of the Sine Function
                                                                   To calculate the derivative of ƒsxd = sin x, for x measured in radians, we combine the lim-
                                                                   its in Example 5a and Theorem 7 in Section 2.4 with the angle sum identity for the sine:
                                                                                             sin sx + hd = sin x cos h + cos x sin h.
                                              Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
                                4100 AWL/Thomas_ch03p147-243  8/19/04  11:16 AM  Page 184
                                                     184                               Chapter 3: Differentiation
                                                                                                                                                                                            If thenƒsxd = sin x,
                                                                                                                                                                                                                           ƒ  ¿sxd = lim ƒsx + hd - ƒsxd
                                                                                                                                                                                                                                                      h:0                                      h
                                                                                                                                                                                                                                            = lim sin sx + hd - sin x                                                                                                                                                           Derivative definition
                                                                                                                                                                                                                                                      h:0                                          h
                                                                                                                                                                                                                                            = lim ssin x cos h + cos x sin hd - sin x                                                                                                                                           Sine angle sum identity
                                                                                                                                                                                                                                                      h:0                                                                 h
                                                                                                                                                                                                                                            = lim sin x scos h - 1d + cos x sin h
                                                                                                                                                                                                                                                      h:0                                                            h
                                                                                                                                                                                                                                            = lim asin x # cos h - 1b + lim acos x # sin hb
                                                                                                                                                                                                                                                      h:0                                                    h                                   h:0                                             h
                                                                                                                                                                                                                                            = sin x # lim cos h - 1 + cos x # lim sin h
                                                                                                                                                                                                                                                                       # h:0                           h#                                                  h:0 h
                                                                                                                                                                                                                                            = sin x 0 + cos x 1                                                                                                                                                                 Example 5(a) and
                                                                                                                                                                                                                                            = cos x.                                                                                                                                                                            Theorem 7, Section 2.4
                                                                                                                                                                                                           The derivative of the sine function is the cosine function:
                                                                                                                                                                                                                                                                                                                d  ssin xd = cos x.
                                                                                                                                                                                                                                                                                                              dx
                                                                                                                                                                                            EXAMPLE 1                                             Derivatives Involving the Sine
                                                                                                                                                                                            (a) y = x2 - sin x:
                                                                                                                                                                                                                                                                                                         dy                                        d
                                                                                                                                                                                                                                                                                                                    = 2x -                                Asin xB                                                  Difference Rule
                                                                                                                                                                                                                                                                                                         dx                                      dx
                                                                                                                                                                                                                                                                                                                  = 2x - cos x.
                                                                                                                                                                                            (b) y = x2 sin x:
                                                                                                                                                                                                                                                                                                   dy                      2 d
                                                                                                                                                                                                                                                                                                               = x                       Asin xB + 2x sin x                                                         Product Rule
                                                                                                                                                                                                                                                                                                   dx                           dx
                                                                                                                                                                                                                                                                                                             = x2 cos x + 2x sin x.
                                                                                                                                                                                            (c)            y = sin x:
                                                                                                                                                                                                                                x
                                                                                                                                                                                                                                                                                                                       x # d  Asin xB - sin x # 1
                                                                                                                                                                                                                                                                                                   dy                            dx
                                                                                                                                                                                                                                                                                                              =                                               2                                                     Quotient Rule
                                                                                                                                                                                                                                                                                                   dx                                                     x
                                                                                                                                                                                                                                                                                                             = x cos x - sin x.
                                                                                                                                                                                                                                                                                                                                            x2
                                                                                                                                                                                            Derivative of the Cosine Function
                                                                                                                                                                                            With the help of the angle sum formula for the cosine,
                                                                                                                                                                                                                                                                           cos sx + hd = cos x cos h - sin x sin h,
                                                                                                                                              Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
           4100 AWL/Thomas_ch03p147-243  8/19/04  11:16 AM  Page 185
                                                                                                                  3.4   Derivatives of Trigonometric Functions      185
                                                                       we have
                                                                                    d  scos xd = lim cossx + hd - cos x                             Derivative definition
                                           y                                       dx             h:0             h
                                          1                y  cos x
                                                                                                = lim scos x cos h - sin x sin hd - cos x            Cosine angle sum
                                                                  x                               h:0                     h                          identity
                              –		
                                         0
                                        –1                                                             cos xscos h - 1d - sin x sin h
                                           y'                                                   = lim                   h
                                                           '                                      h:0
                                                          y  –sin x
                                          1                                                                  # cos h - 1                 # sin h
                                                                  x                             = lim cos x         h      - lim sin x      h
                              –		 h:0                                                                                         h:0
                                         0
                                        –1                                                              #      cos h - 1            #     sin h
                                                                                                = cos x   lim       h      - sin x   lim  h
                         FIGURE 3.23     The curve y¿=-sin x as                                         # h:0        #               h:0
                         the graph of the slopes of the tangents to                             = cos x 0 - sin x 1                                  Example 5(a) and
                         the curve y = cos x.                                                   =-sin x.                                             Theorem 7, Section 2.4
                                                                            The derivative of the cosine function is the negative of the sine function:
                                                                                                              d  scos xd =-sin x
                                                                                                             dx
                                                                       Figure 3.23 shows a way to visualize this result.
                                                                       EXAMPLE 2          Derivatives Involving the Cosine
                                                                       (a) y = 5x + cos x:
                                                                                                  dy      d           d
                                                                                                      =      s5xd +      Acos xB                     Sum Rule
                                                                                                  dx     dx          dx
                                                                                                       = 5 - sin x.
                                                                       (b) y = sin x cos x:
                                                                                            dy           d                     d
                                                                                                = sin x     Acos xB + cos x       Asin xB            Product Rule
                                                                                            dx          dx                    dx
                                                                                                 = sin xs-sin xd + cos xscos xd
                                                                                                 = cos2 x - sin2 x.
                                                                       (c)  y =     cos x   :
                                                                                  1 - sin x
                                                                                            A1 - sin xB  d  Acos xB - cos x  d  A1 - sin xB
                                                                                     dy                  dx                    dx
                                                                                         =                                 2                         Quotient Rule
                                                                                     dx                        s1 - sin xd
                                                                                         = s1 - sin xds-sin xd - cos xs0 - cos xd
                                                                                                                        2
                                                                                                           s1 - sin xd
                                                                                             1 - sin x                                                 2       2
                                                                                         =              2                                            sin  x + cos  x = 1
                                                                                            s1 - sin xd
                                                                                         =      1     .
                                                                                            1 - sin x
                                                 Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
          4100 AWL/Thomas_ch03p147-243  8/19/04  11:16 AM  Page 186
                 186        Chapter 3: Differentiation
                                                            Simple Harmonic Motion
                                                            The motion of a body bobbing freely up and down on the end of a spring or bungee cord is
                                                            an example of simple harmonic motion. The next example describes a case in which there
                                        –5                  are no opposing forces such as friction or buoyancy to slow the motion down.
                                                            EXAMPLE 3         Motion on a Spring
                                        0    Rest
                                             position       A body hanging from a spring (Figure 3.24) is stretched 5 units beyond its rest position
                                                            and released at time        to bob up and down. Its position at any later time t is
                                                                                 t = 0
                                        5    Position at                                            s = 5 cos t.
                                             t  0
                                      s                     What are its velocity and acceleration at time t?
                 FIGURE 3.24    A body hanging from         Solution    We have
                 avertical spring and then displaced             Position:           s = 5 cos t
                 oscillates above and below its rest position.   Velocity:           y = ds = d s5 cos td =-5 sin t
                 Its motion is described by trigonometric                                 dt     dt
                 functions (Example 3).                                                   dy     d
                                                                 Acceleration:       a = dt = dt s-5 sin td =-5 cos t.
                   s, y                                          Notice how much we can learn from these equations:
                                                            1.   As time passes, the weight moves down and up between s =-5 and s = 5 on the
                         y  –5 sin t          s  5 cos t       s-axis. The amplitude of the motion is 5. The period of the motion is 2p.
                                                            2.   Thevelocityy =-5 sin tattainsitsgreatestmagnitude,5,whencos t = 0,asthe graphs
                                                                 show in Figure 3.25. Hence, the speed of the weight, ƒ y ƒ = 5 ƒ  sin t ƒ , is greatest when
                                                        t        cos t = 0,thatis,whens = 0(the rest position). The speed of the weight is zero when
                   0     3 2 5                              sin t = 0.Thisoccurswhens = 5 cos t =;5,attheendpointsoftheintervalofmotion.
                         2            2         2
                                                            3.   The acceleration value is always the exact opposite of the position value. When the
                                                                 weight is above the rest position, gravity is pulling it back down; when the weight is
                                                                 below the rest position, the spring is pulling it back up.
                                                            4.   The acceleration,                 is zero only at the rest position, where          and
                 FIGURE 3.25    The graphs of the position                         a =-5 cos t,                                           cos t = 0
                                                                 the force of gravity and the force from the spring offset each other. When the weight is
                 and velocity of the body in Example 3.          anywhere else, the two forces are unequal and acceleration is nonzero. The accelera-
                                                                 tion is greatest in magnitude at the points farthest from the rest position, where
                                                                 cos t =;1.
                                                            EXAMPLE 4         Jerk
                                                            The jerk of the simple harmonic motion in Example 3 is
                                                                                         j = da = d s-5 cos td = 5 sin t.
                                                                                              dt    dt
                                                            It has its greatest magnitude when               not at the extremes of the displacement but
                                                                                                sin t =;1,
                                                            at the rest position, where the acceleration changes direction and sign.
                                                            Derivatives of the Other Basic Trigonometric Functions
                                                            Because sin x and cos x are differentiable functions of x, the related functions
                                                                 tan x = sin x , cot x = cos x , sec x =            1  , and csc x = 1
                                                                         cos x                sin x               cos x                        sin x
                                              Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
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...Awl thomas chp am page derivatives of trigonometric functions many the phenomena we want information about are approximately periodic electro magnetic fields heart rhythms tides weather sines and cosines play a key role in describing changes this section shows how to differentiate six ba sic derivative sine function calculate sxd sin x for measured radians combine lim its example theorem with angle sum identity sx hd cos h copyright pearson education inc publishing as addison wesley chapter differentiation if then definition ssin scos d asin b acos hb is cosine xd dx involving y dy xb difference rule...

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