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3.8Derivatives of Inverse Trigonometric Functions
Quick Review
Slide 3- 2
In Exercises 1-5, give the and of the function,
and evaluate the function at 1.
domain range
x
1
1
1
1
1
1. sin
2. cos
3. tan
4. sec
5. tan tan
y x
y x
y x
y x
y x
Quick Review
Slide 3- 3
In Exercises 6-10, find the inverse of the given function.
3
6. 3 8
7. 5
88.
3 29.
10. arctan 3
y x
y x
yxx
yx
xy
Quick Review Solutions
Slide 3- 4
In Exercises 1-5, give the and of the function,
and evaluate the function at 1.
domain range
x
1
1
1
1
1. sin
2. cos
3. tan
Domain: 1,1 Range: - , At 1:2 2 2
Domain: 1,1 Range: 0, At 1:0
Domain:All R
4.
eals Range: - , At 1:2 2 4
Domain: , 1 1,
Range: 0, ,2 2
sec
y x
y x
y x
y x
∪
∪
1
At 1:0
Domain:All Reals Rang5. e:Al tan l Reals At n 11a :ty x
Quick Review Solutions
Slide 3- 5
1
1 3
1
1
1
3
6. 3 8
7. 5
88.
3 29.
8
3
5
8
2
3
3tan ,
10. arct23 2
an
xf x
f x x
f xx
f xx
f
y x
y x
yxx
y
xx
x
y x x
In Exercises 6-10, find the inverse of the given function.
What you’ll learn about
Derivatives of Inverse Functions
Derivatives of the Arcsine
Derivatives of the Arctangent
Derivatives of the Arcsecant
Derivatives of the Other Three
… and why
The relationship between the graph of a function and its inverse
allows us to see the relationship between their derivatives.
Slide 3- 6
Derivatives of Inverse Functions
If f is differentiable at every point of an interval I and dy
dxis never zero on I , then f has an inverse and f 1 is differentiable
at every point on the interval f I .
Slide 3- 7
Derivative of the Arcsine
If u is a differentiable function of x with u 1, we apply the
Chain Rule to get
d
dxsin 1 u
1
1 u2
du
dx, u 1.
Slide 3- 8
Let f(x) = sin x and g(x) = sin-1 x to verify the formula for the derivative of sin-1 x.
Example Derivative of the Arcsine
If ysin 1 8x2 , find
dy
dx.
Slide 3- 10
Example Derivative of the Arcsine
If ysin 1(1 t), find
dy
dx.
Slide 3- 11
Derivative of the Arctangent
The derivative is defined for all real numbers.
If u is a differentiable function of x, we apply the
Chain Rule to get
d
dxtan 1 u
1
1u2
du
dx.
Slide 3- 12
y = tan-1 (4x)
Determine
dy
dx.
y = x tan-1x
Determine
dy
dx.
Derivative of the Arcsecant
If u is a differentiable function of x with u 1, we have the
formula
d
dxsec 1 u
1
u u2 1
du
dx, u 1.
Slide 3- 15
Example Derivative of the Arcsecant
1Given sec 3 4 , find .dy
y xdx
Slide 3- 16
A particle moves along the x – axis so that its position at any time t ≥ 0 is given by x(t). Find the velocity at the indicated value of t.
x(t) sin 1 t
4
, t 4
Assignment 3.8.1
page 170,
# 3 – 11 odds
Inverse Function – Inverse Cofunction
Identities
1 1
1 1
1 1
cos sin2
cot tan2
csc sec2
x x
x x
x x
Slide 3- 19
Determine
dy
dx if y cos 1 x.
Derivatives of Inverse Trig Functions
Function
arcsin x
arccos x
arctan x
arcsec x
Derivative
2
1
1 x 2
1
1 x
2
1
1 x2
1
1x x
Example Derivative of the Arccotangent
Slide 3- 22
1 2Find the derivative of cot .y x
Calculator Conversion Identities
1 1
1 1
1 1
1sec cos
cot tan2
1csc sin
xx
x x
xx
Slide 3- 23
Determine the derivative of y with respect to the variable.
y cos 1 1
x
Determine the derivative of y with respect to the variable.
y sec 1 5s
Determine the derivative of y with respect to the variable.
y csc 1 x
2
Determine the derivative of y with respect to the variable.
y s2 1 sec 1 s
Find an equation for the tangent to the graph of y at the indicated point.
y tan 1 x, x 2
Find an equation for the tangent to the graph of y at the indicated point.
y cos 1 x
4
, x 5
Let f(x) = cos x + 3x
Show that f(x) has a differentiable inverse.
Let f(x) = cos x + 3x
Determine f(0) and f’(0).
Let f(x) = cos x + 3x
Determine f-1(1) and f-1(1).
y = cot-1 x
Determine the right end behavior model.
y = cot-1 x
Determine the left end behavior model.
y = cot-1 x
Does the function have any horizontal tangents?
Assignment 3.8.2
pages 170 – 171,
# 1, 13 – 29 odds, 32 and 41 – 45 odds
