Derivatives of Inverse Trigonometric Functions
27. (a) Give three di↵erent domain intervals of length ⇡ where the sine function is one-
to-one.
(b) Sketch a graph of y = sin x on the restricted domain [�⇡2 ,
⇡
2 ]. Give the range in
interval notation.
(c) Define the inverse sine (arcsine) function by writing y = arcsin x if and only if
x = sin y, where y 2 [�⇡2 ,
⇡
2 ] and x 2 [�1, 1]. Sketch a graph of the arcsine
function. Give the domain and range of the arcsine function. How is this graph
elated to the graph of the sine?
28. Let y = arcsin x.
(a) Solve for x.
(b) Take the derivative of both sides of your equation from part (a). Employ the
chain rule every time you need to apply the ddx to a y-variable.
(c) Solve fo
dy
dx
. Use your equation from part (a) to write your answer as a function
of x.
(d) Use your result to complete the following Theorem.
Theorem 10 (Derivative of the Inverse Sine)
d
dx
arcsin x =
29. (a) Sketch a graph of y = tan x on the restricted domain (�⇡2 ,
⇡
2 ). Give the range in
interval notation.
(b) Define the inverse tangent (arctangent) function by writing y = arctan x if and
only if x = tan y, where y 2 (�⇡2 ,
⇡
2 ). Sketch a graph of the arctangent function.
Give the domain and range of the arctangent function. How is this graph related
to the graph of the tangent?
30. Let y = arctan x.
(a) Solve for x.
(b) Take the derivative of both sides of your equation from part (a). Employ the
chain rule every time you need to apply the ddx to a y-variable.
(c) Solve fo
dy
dx
. Use your equation from part (a) to write your answer as a function
of x.
(d) Use your result to complete the following Theorem.
Theorem 11 (Derivative of the Inverse Tangent)
d
dx
arctan x =
31. (a) Sketch an accurate graph of the cosine on the domain (�2⇡, 2⇡). Use this to
sketch an accurate graph of the secant function on the same set of axes.
(b) Identify at least three domains of length ⇡ where the secant is one-to-one.
32. (a) Sketch y = secx on a restricted domain of length ⇡ where the function is one-to-
one. Choose such a domain with angles in Quadrants 1 and 3 only.
(b) Sketch a graph of the inverse secant function defined using this domain restriction.
(c) Give the domain and range of the inverse secant according to this definition.
(d) Give the open intervals where the inverse secant is increasing or decreasing. What
does this tell you about the derivative of the inverse secant?
(e) Show that
dy
dx
=
1
x
p
x2 � 1
.
33. (a) Sketch y = secx on a restricted domain of length ⇡ where the function is one-to-
one. Choose such a domain with angles in Quadrants 1 and 2 only.
(b) Sketch a graph of the inverse secant function defined using this domain restriction.
Give the domain and range of the inverse secant according to this definition.
(c) Give the open intervals where the arcsecant is increasing or decreasing. What
does this tell you about the derivative of the inverse secant?
(d) Show that
dy
dx
=
1
|x|
p
x2 � 1
.
Theorem 12 (Derivative of the Inverse Secant)
d
dx
sec�1 x =
1
|x|
p
x2 � 1