Problem 1.   If  y = arcsin x, show:

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Begin:

	y	=	arcsin x
implies
	1)   sin y	=	x.
Therefore, according to the Pythagorean identity a':
	cos y	=
=
according to line 1).

Problem 2.   If  y = arcsec x, show:

Begin:

	y	=	arcsec x
implies
	sec y	=	x.
Therefore, according to the Pythagorean identity b:
	tan y	=
		=

The derivative of y = arcsin x

Here is the proof:

	y	=	arcsin x
implies
	sin y	=	x.
Therefore, on taking the derivative with respect to x:
		=	1;
		=
		=

The derivative of y = arccos x

The derivative of  arccos x  is the negative of the derivative
of  arcsin x.  That will be true for each pair of cofunctions.

The derivative of  arccot x  will be the negative
of the derivative of  arctan x.

The derivative of  arccsc x  will be the negative
of the derivative of  arcsec x.

For, beginning with arccos x:

The angle whose cosine is x is the complement
of the angle whose sine is x.

cos α = sin β. Therefore, if α is the angle whose cosine is x, it is the complement of β, the angle whose sine is x.

And similarly for each pair of cofunctions.

Problem 4.   Calculate these derivatives.

a)

d  dx

arccos

x a

=

b)

d  dx

x arccos 2x

=

The derivative of y = arctan x

First,

y = arctan x   implies   tan y = x.

The derivative of y = arcsec x

Again,

y = arcsec x   implies   sec y = x.