Chapter 2: Differentiation
Section 2.8: The Inverse Trig Functions and Their Derivatives
Essentials
Introduction
Table 2.8.1 contains a graph of the principal branch for each of the six trig functions, and the graph of the corresponding inverse function.
Principal Branch
Principal Domain
Inverse Function
−π2≤x≤π2
0≤x≤π
−π2≤x≤π2
Table 2.8.1 Principal branches of the trig functions, and their inverse functions
Table 2.8.2 lists the derivatives of the six inverse trigonometric functions.
Function
Maple's Derivative
Textbook Derivative
arcsine
ⅆⅆ x arcsinx = 11−x2
11−x2, −1<x<1
arccosine
ⅆⅆ x arccosx = −11−x2
−11−x2, −1<x<1
arctangent
ⅆⅆ x arctanx = 11+x2
11+x2
arccotangent
ⅆⅆ x arccotx = −11+x2
−11+x2
arcsecant
ⅆⅆ x arcsecx = 1x21−1x2
1x x2−1, x>1
arccosecant
ⅆⅆ x arccscx = −1x21−1x2
−1x x2−1, x>1
Table 2.8.2 Derivatives of the six inverse trigonometric functions
Maple's differentiation formulas are correct for x complex, but in the typical calculus text, the formulas are stated for x real. That is why the textbook formulas have restrictions and vary slightly from the Maple form.
The second column in the table uses the formal name of each inverse function. However, in 2D math mode, Maple understands the usage sin−1x for arcsinx, etc.
Graphing an Inverse Function
The graph of f−1, the functional inverse of the function f, is the set of ordered pairs in which the abscissas and ordinates have been reversed. This interchange of the values of x and fx can be accomplished graphically by drawing the curve defined parametrically by x=ft,y=t, that is, by graphing the points fx,x.
See Example 2.8.1.
Determining the Principal Branch
The curve drawn parametrically by x=f−1t,y=t, that is, the graph of the points f−1x,x, is the graph of the inverse of the inverse, and is therefore the graph of f. This device will work to determine the principal branch of each of the trig functions.
See Example 2.8.2.
Graphs of the Inverse Trig Functions and their Derivatives
Select Functionarcsinearccosinearctangentarccotangentarcsecantarccosecant
Figure 2.8.1 Graph of inverse trig function (red) and its derivative (blue)
Figure 2.8.1 can contain a graph of one of the inverse trig functions (drawn in red) and its derivative (drawn in blue). Simply select a function from the drop-down box below the space where the graph is to appear.
Alternatively, select an inverse trig function from the drop-down box in Table 2.8.3, then press the button "Launch Tutor" to launch the Derivatives tutor with the corresponding inverse function embedded.
arcsinearccosinearctangentarccotangentarcsecantarccosecant
Table 2.8.3 Access the Derivatives tutor
Précis
The graphs of the principal branches of the trig functions and the graphs of the corresponding inverse trig functions are given in Table 2.8.1.
The derivatives of the inverse trig functions are listed in Table 2.8.2.
Examples
Example 2.8.1
Graph fx,x for the function fx=x2,0≤x≤2.
Example 2.8.2
Graph the inverse of arcsinx by graphing parametrically the points arcsinx,x.
Example 2.8.3
Evaluate ⅆⅆx tan−1x.
Example 2.8.4
Evaluate ⅆⅆx sin−1x2.
Example 2.8.5
Evaluate ⅆⅆx sec−11/x.
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