3.8 derivatives of inverse trigonometric functions
TRANSCRIPT
![Page 1: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/1.jpg)
S
3.8Derivatives of Inverse Trigonometric Functions
![Page 2: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/2.jpg)
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
![Page 3: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/3.jpg)
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
![Page 4: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/4.jpg)
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
![Page 5: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/5.jpg)
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.
![Page 6: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/6.jpg)
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
![Page 7: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/7.jpg)
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
![Page 8: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/8.jpg)
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
![Page 9: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/9.jpg)
Let f(x) = sin x and g(x) = sin-1 x to verify the formula for the derivative of sin-1 x.
![Page 10: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/10.jpg)
Example Derivative of the Arcsine
If ysin 1 8x2 , find
dy
dx.
Slide 3- 10
![Page 11: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/11.jpg)
Example Derivative of the Arcsine
If ysin 1(1 t), find
dy
dx.
Slide 3- 11
![Page 12: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/12.jpg)
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
![Page 13: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/13.jpg)
y = tan-1 (4x)
Determine
dy
dx.
![Page 14: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/14.jpg)
y = x tan-1x
Determine
dy
dx.
![Page 15: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/15.jpg)
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
![Page 16: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/16.jpg)
Example Derivative of the Arcsecant
1Given sec 3 4 , find .dy
y xdx
Slide 3- 16
![Page 17: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/17.jpg)
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
![Page 18: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/18.jpg)
Assignment 3.8.1
page 170,
# 3 – 11 odds
![Page 19: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/19.jpg)
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
![Page 20: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/20.jpg)
Determine
dy
dx if y cos 1 x.
![Page 21: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/21.jpg)
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
![Page 22: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/22.jpg)
Example Derivative of the Arccotangent
Slide 3- 22
1 2Find the derivative of cot .y x
![Page 23: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/23.jpg)
Calculator Conversion Identities
1 1
1 1
1 1
1sec cos
cot tan2
1csc sin
xx
x x
xx
Slide 3- 23
![Page 24: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/24.jpg)
Determine the derivative of y with respect to the variable.
y cos 1 1
x
![Page 25: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/25.jpg)
Determine the derivative of y with respect to the variable.
y sec 1 5s
![Page 26: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/26.jpg)
Determine the derivative of y with respect to the variable.
y csc 1 x
2
![Page 27: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/27.jpg)
![Page 28: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/28.jpg)
Determine the derivative of y with respect to the variable.
y s2 1 sec 1 s
![Page 29: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/29.jpg)
Find an equation for the tangent to the graph of y at the indicated point.
y tan 1 x, x 2
![Page 30: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/30.jpg)
Find an equation for the tangent to the graph of y at the indicated point.
y cos 1 x
4
, x 5
![Page 31: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/31.jpg)
![Page 32: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/32.jpg)
Let f(x) = cos x + 3x
Show that f(x) has a differentiable inverse.
![Page 33: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/33.jpg)
Let f(x) = cos x + 3x
Determine f(0) and f’(0).
![Page 34: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/34.jpg)
Let f(x) = cos x + 3x
Determine f-1(1) and f-1(1).
![Page 35: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/35.jpg)
y = cot-1 x
Determine the right end behavior model.
![Page 36: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/36.jpg)
y = cot-1 x
Determine the left end behavior model.
![Page 37: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/37.jpg)
y = cot-1 x
Does the function have any horizontal tangents?
![Page 38: 3.8 Derivatives of Inverse Trigonometric Functions](https://reader038.vdocument.in/reader038/viewer/2022103006/56649e875503460f94b8b3de/html5/thumbnails/38.jpg)
Assignment 3.8.2
pages 170 – 171,
# 1, 13 – 29 odds, 32 and 41 – 45 odds