Section 33–3
◆
979
Derivatives of the Inverse Trigonometric Functions
38. The range x of a projectile fired at an angle with the horizontal at a velocity v (Fig. 33–12)
is x (v2/g) sin 2, where g is the acceleration due to gravity. Find for the maximum
range.
θ
x
FIGURE 33–12
39. A force F (Fig. 33–13) pulls the weight along a horizontal surface. If f is the coefficient of
friction, then
fW
W
F = f sin cos Find for a minimum force when f 0.60.
FIGURE 33–13
40. A 6.00-m-long steel girder is dragged along a corridor 3.00 m wide and then around a
corner into another corridor at right angles to the first (Fig. 33–14). Neglecting the thickness of the girder, what must the width of the second corridor be to allow the girder to turn
the corner?
Girder
FIGURE 33–14
Top view of a girder dragged along a corridor.
41. If the girder in Fig. 33–14 is to be dragged from a 3.20-m-wide corridor into another
1.35-m-wide corridor, find the length of the longest girder that will fit around the corner.
(Neglect the thickness of the girder.)
33–3 Derivatives of the Inverse Trigonometric Functions
Derivative of Arcsin u
We now seek the derivative of y sin 1 u where y is some angle whose sine is u, as in
Fig. 33–15, whose value we restrict to the range /2 to /2. We can then write
sin y u
1
u
y
1 − u2
FIGURE 33–15
Taking the derivative yields
dy du
cos y = dx dx
so
dy
1 du
cos y dx
= dx
(1)
F
980
Chapter 33
◆
Derivatives of Trigonometric, Logarithmic, and Exponential Functions
However, from Eq. 164,
cos2 y = 1 sin2 y
cos y = 1 sin2 y
But since y is restricted to values between /2 and /2, cos y cannot be negative. So
cos y = 1 sin2 y = 1 u2
Substituting into Eq. (1), we have the following equation:
d(Sin1 u)
du
1 u2 dx
1
Derivative of
the Arcsin
dx
356
1 < u < 1
Example 13: If y Sin1 3x, then
◆◆◆
1
y 2 (3)
1 (3x)
3
2
1 9x
◆◆◆
Derivatives of Arccos, Arctan, Arccot, Arcsec, and Arccsc
The rules for taking derivatives of the remaining inverse trigonometric functions, and the Arcsin
as well, are as follows:
Try to derive one or more of
these equations. Follow steps
similar to those we used for the
derivative of Arcsin.
d(Sin1 u)
du
1 u dx
1 < u < 1
1
dx
2
d(Cos1 u)
1
du
dx
1 u dx
1 < u < 1
2
d (Tan1 u)
Derivatives
of the Inverse
Trigonometric
Functions
d (Cot1 u)
358
1 du
1 u2 dx
359
dx
d(Sec1 u)
du
uu2 1 dx
u > 1
1
dx
d(Csc1 u)
du
u u 1 dx
2
u > 1
360
1
dx
357
du
1
1 u2 dx
dx
356
361
Section 33–4
◆
981
Derivatives of Logarithmic Functions
◆◆◆
Example 14:
(a) If y Cot1 (x2 1), then
1
y = (2x)
1 (x2 1)2
2x
2 2x2 x4
(b) If y Cos1 1 x , then
1
1
y = 1
(1
x)
2
1 x
1
= 2
2x x
Exercise 3
◆
◆◆◆
Derivatives of the Inverse Trigonometric Functions
Find the derivative.
1. y x Sin1 x
x
2. y Sin1 a
x
3. y Cos1 a
4. y Tan1(sec x tan x)
sin x cos x
5. y Sin1 2
2ax x2
6. y 2ax x2 a Cos1 a
7. y t2 Cos1 t
9. y Arctan(1 2x)
8. y Arcsin 2x
10. y Arccot(2x 5)2
x
11. y Arccot a
1
12. y Arcsec x
13. y Arccsc 2x
14. y Arcsin x
t
15. y t2 Arcsin 2
a
1
17. y Sec 2
a x2
x
16. y Sin1 2
1 x
Find the slope of the tangent to each curve.
1
18. y x Arcsin x at x 2
20. y x2 Arccsc x at x 2
Arctan x
19. y at x 1
x
x
x Arccot at x 4
21. y 4
22. Find the equations of the tangents to the curve y Arctan x having a slope of 14 .
33–4 Derivatives of Logarithmic Functions
Derivative of logb u
Let us now use the delta method again to find the derivative of the logarithmic function y logb u. We first let u take on an increment u and y an increment y.