AP CALCULUS BC
Section 5.6 (day 1): Differentiation of Inverse Trigonometric Functions, pg. 371
DO NOW:
NONE OF THE SIX TRIGONOMETRIC FUNCTIONS HAS AN INVERSE FUNCTION.
Unless we impose a restriction to each of the domains in order to make each of
the six trigonometric functions one-to-one.
For each trigonometric function, write the restricted domain (in order to make the
function one-to-one ) and range:
y = sin( x)
Restricted Domain: _______________ Range: ________________
y = cos( x)
Restricted Domain: _______________ Range: ________________
y = tan( x)
Restricted Domain: _______________ Range: ________________
y = csc( x)
Restricted Domain: _______________ Range: ________________
y = sec( x)
Restricted Domain: _______________ Range: ________________
y = cot( x)
Restricted Domain: _______________ Range: ________________
To find the derivative of the inverse function y = arcsin( x ) we’ll use our knowledge of
implicit differentiation and the trigonometry of right triangles.
Find
dy
for y = arcsin( x )
dx
To find the derivative of the inverse function y = arccos( x ) we’ll use our knowledge of
implicit differentiation and the trigonometry of right triangles.
Find
dy
for y = arccos( x )
dx
To find the derivative of the inverse function y = arctan( x ) we’ll use our knowledge of
implicit differentiation and the trigonometry of right triangles.
Find
dy
for y = arctan( x )
dx
To find the derivative of the inverse function y = arccot( x ) we’ll use our knowledge of
implicit differentiation and the trigonometry of right triangles.
Find
dy
for y = arccot( x )
dx
To find the derivative of the inverse function y = arcsec( x ) we’ll use our knowledge of
implicit differentiation and the trigonometry of right triangles.
Find
dy
for y = arcsec( x )
dx
To find the derivative of the inverse function y = arccsc( x ) we’ll use our knowledge of
implicit differentiation and the trigonometry of right triangles.
Find
dy
for y = arccsc( x )
dx
SAMPLE PROBLEMS:
Differentiate the following functions:
1.
y = 12arccos(−2 x)
3.
y = x(arctan( x))
5.
1
y = arccot   − arctan x
x
2
HOMEWORK: pg. 378: 41 - 66
2.
y = arcsin x + x 1 − x 2
4.
y = arcsin ( x
(
6.
3
))
4
y = x arcsin x + 1 − x 2