TRIGONOMETRIC FUNCTIONS AND THEIR DERIVATIVES
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10.14
Calculate
10.15
Calculate
10.16
Calculate
10.17
Using the A-definition, calculate
Thus,
I By the identity sin (u + v) = sin ucos v + cos wsin v, sin (x + Ax) = sin x cos (A*) + cos x sin (Ax). Hence,
sin (x + Ax) - sin ;c = sin x[cos (Ax) — 1] + cos x sin (Ax), and
Here, we have used
(Problem 10.16).
10.18
Calculate
(cos x) from the known derivative of sin x
10.19
Calculate
sin 3x is a composite function of 3x and the sine function. By the chain rule and the fact that
10.20
Calculate
Hence, by the chain rule,
10.21
Find
10.22
Find an equation of the tangent line to the graph of y = sin
2 x at the point where x = ir/3.
The slope of the tangent line is the derivative y'. By the chain rule, since sin
2 x = (sinx)
2 , y'=2(sinx)and
When * = 7r/3, sinx = V5/2
At the point where x = if 13, y = (V5/2)
2 = i. So a point-slope equation of the tangent line is y — \ =
and
[chain
By the identity cos x = sin
rule] = sin x • (-1)
cos
2 x = (cos x)
2 .
-2 sin x cos x = -sin 2x.
By the chain rule,
(sin x) = 2 sin x cos x.
cosx=i. So /=2-V3/2-i=V§/2.
(V3/2)(Ar--n-/3).