5128_Ch03_pp098-184

Section 3.8 Derivatives of Inverse Trigonometric Functions 167
EXAMPLE 1
Applying the Formula
d
sin d x
1 x 2 1 d
1 x 2
2 • x d x
2 2x
1 x 4
Now try Exercise 3.
Derivative of the Arctangent
Although the function y sin 1 x has a rather narrow domain of 1, 1, the function
y tan 1 x is defined for all real numbers, and is differentiable for all real numbers, as
we will now see. The differentiation proceeds exactly as with the arcsine function.
y tan 1 x
tan y x
d d
tan y x
d x d x
sec 2 dy
y 1 d x
dy 1
d x sec 2 y
Inverse function relationship
Implicit differentiation
1
1 tan y 2 Trig identity: sec 2 y 1 tan 2 y
1
1 x 2
The derivative is defined for all real numbers. If u is a differentiable function of x, we get
the Chain Rule form:
d
tan d x
1 1
u 1 u 2 d u
.
dx
EXAMPLE 2 A Moving Particle
A particle moves along the x-axis so that its position at any time t 0 is xt tan 1 t.
What is the velocity of the particle when t 16?
SOLUTION
d
vt tan d t
1 1 d 1 1
t • t •
1 t 2 d t 1 t 2 t
1 1 1
When t 16, the velocity is v16 • Ṅow 1 16 2 16 1 36
try Exercise 11.
Derivative of the Arcsecant
We find the derivative of y sec 1 x, x 1, beginning as we did with the other inverse
trigonometric functions.
y sec 1 x
sec y x
Inverse function relationship
d d
sec y x
d x d x
dy
sec y tan y 1 d x
dy 1
d x sec y tan y
Since ⏐x⏐ 1, y lies in (0, p2) (p2, p)
and sec y tan y 0.