James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.3 Derivatives of Trigonometric Functions 193
Figure 3 shows the graphs of the
function of Example 1 and its derivative.
Notice that y9 − 0 whenever y has
a horizontal tangent.
5
ExamplE 1 Differentiate y − x 2 sin x.
SOLUtion Using the Product Rule and Formula 4, we have
dy
dx − x 2 d dx ssin xd 1 sin x d dx sx 2 d
− x 2 cos x 1 2x sin x
■
_4 4
yª
y
Using the same methods as in the proof of Formula 4, one can prove (see Exercise
20) that
5
d
scos xd − 2sin x
dx
FIGURE 3
_5
The tangent function can also be differentiated by using the definition of a derivative,
but it is easier to use the Quotient Rule together with Formulas 4 and 5:
d
S
dx stan xd − d sin x
dx cos xD
cos x d dx ssin xd 2 sin x d scos xd
dx
−
cos 2 x
−
cos x ? cos x 2 sin x s2sin xd
cos 2 x
− cos2 x 1 sin 2 x
cos 2 x
− 1
cos 2 x − sec2 x
6
d
dx stan xd − sec2 x
The derivatives of the remaining trigonometric functions, csc, sec, and cot, can also
be found easily using the Quotient Rule (see Exercises 17–19). We collect all the differentiation
formulas for trigonometric functions in the following table. Remember that
they are valid only when x is measured in radians.
When you memorize this table, it is
helpful to notice that the minus signs
go with the derivatives of the “cofunctions,”
that is, cosine, cosecant, and
cotangent.
Derivatives of Trigonometric Functions
d
d
ssin xd − cos x
dx
d
d
scos xd − 2sin x
dx
d
dx stan xd − sec2 x
scsc xd − 2csc x cot x
dx
ssec xd − sec x tan x
dx
d
dx scot xd − 2csc2 x
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.