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

200 Chapter 3 Differentiation Rules
ExamplE 2 Differentiate (a) y − sinsx 2 d and (b) y − sin 2 x.
SOLUTION
(a) If y − sinsx 2 d, then the outer function is the sine function and the inner function is
the squaring function, so the Chain Rule gives
dy
dx − d
dx
sin sx 2 d − cos sx 2 d ? 2x
outer
function
− 2x cossx 2 d
evaluated
at inner
function
derivative
of outer
function
evaluated
at inner
function
derivative
of inner
function
(b) Note that sin 2 x − ssin xd 2 . Here the outer function is the squaring function and the
inner function is the sine 7et0304note2
function. So
dy 01/13/10
dx − d
dx MasterID: ssin xd2 01593 − 2 ? ssin xd ? cos x
inner
function
inner derivative derivative evaluated
function of outer of outer at inner
function function
evaluatedderivative
at inner of inner
function function
derivative
of inner
function
See Reference Page 2 or Appendix D.
The answer can be left as 2 sin x cos x or written as sin 2x (by a trigonometric identity
known as the double-angle formula).
■
7et0304note3 7et0304note3
In Example 2(a) we combined 01/13/10 the 01/13/10 Chain Rule with the rule for differentiating the sine
function. In general, if y −MasterID: sin u, MasterID: where 01594 u is a differentiable 01594 function of x, then, by the
Chain Rule,
dy
dx − dy du du
− cos u
du dx dx
Thus
d
du
ssin ud − cos u
dx dx
In a similar fashion, all of the formulas for differentiating trigonometric functions can
be combined with the Chain Rule.
Let’s make explicit the special case of the Chain Rule where the outer function f is
a power function. If y − ftsxdg n , then we can write y − f sud − u n where u − tsxd. By
using the Chain Rule and then the Power Rule, we get
dy
dx − dy
du
du
dx
− nu
n21
du
dx − nftsxdgn21 t9sxd
4 The Power Rule Combined with the Chain Rule If n is any real number
and u − tsxd is differentiable, then
Alternatively,
d
dx su n n21
du
d − nu
dx
d
dx ftsxdgn − nftsxdg n21 ? t9sxd
Notice that the derivative in Example 1 could be calculated by taking n − 1 2 in Rule 4.
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