Download 3.8 Derivatives of Inverse Trig Functions

AP Calculus BC
3.8 Derivatives of Inverse Trigonometric Functions
Objective: able to calculate derivatives of the six inverse trigonometric functions.
Recall from Section 1.6
Trigonometric Functions
Inverse Trigonometric Functions
Derivatives of Inverse Functions
dy
dx
is differentiable
If f is differentiable at every point of an interval I and
is never zero on I , then f has an inverse and f −1
at every point on the interval f ( I ) .
How are the derivatives of a function and its inverse related?
1. Find the derivative of arcsin.
2. Find
if
=
√2
.
3. Find the derivative of arctangent.
4. A particle moves along the x-axis so that its position at any time
tan
. Find the velocity at t = 2.
5. Find the derivative of arcsecant.
6. Find
if
=
.
≥ 0 is given by
=
Derivatives of the Other Three
We could use the same techniques to find the derivatives of the other three inverse trigonometric functions:
arccosine, arccotangent, and arccosecant, but it is much easier to think of the following identities.
Inverse Function-Inverse Cofunction Identities
cos −1 x =
cot −1 x =
csc −1 x =
π
2
π
2
π
2
Calculator Conversion Identities
− sin −1 x
1
sec −1 x = cos −1  
x
− tan −1 x
cot −1 x =
− sec −1 x
1
csc −1 x = sin −1  
x
π
2
− tan −1 x
7. Using the identities above, find the derivative of arccosine, arccotangent, and arccosecant.
8. Find the equation of the tangent line to the graph of
=
at
x
= 2.
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