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Chapter 3 | Derivatives
d dx (cos
x ) =−sinx.
Theorem 3.8: The Derivatives of sin x and cos x The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine. (3.11) d dx (sin x ) =cos x (3.12) d dx (cos x ) =−sin x
Proof Because the proofs for d
x ) =cos x and d
dx (sin
dx (cos
x ) =−sin x use similar techniques, we provide only the proof for
d dx (sin x ) =cos x . Before beginning, recall two important trigonometric limits we learned in Introduction to Limits : lim h →0 sin h h =1and lim h →0 cos h −1 h =0. The graphs of y = (sin h ) h and y = (cos h −1) h are shown in Figure 3.26 .
Figure 3.26 These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.
We also recall the following trigonometric identity for the sine of the sum of two angles: sin( x + h ) = sin x cos h +cos x sin h . Now that we have gathered all the necessary equations and identities, we proceed with the proof. d dx sin x = lim h →0 sin( x + h )−sin x h Apply the definition of the derivative. = lim h →0 sin x cos h +cos x sin h −sin x h
Use trig identity for the sine of the sum of two angles.
⎛ ⎝ sin x cos h −sin x h
⎞ ⎠ Regroup.
x sin h h
= lim
+ cos
h →0
⎛ ⎝ sin x
⎛ ⎝ cos h −1 h
⎞ ⎠ +cos x
⎛ ⎝ sin h h
⎞ ⎠ ⎞ ⎠ Factor outsin x andcos x .
= lim
h →0
= sin x ·0+cos x ·1
Apply trig limit formulas.
=cos x
Simplify.
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