Mathematics

Problem

For

0<x<1

, which of the following correctly describes the sign of

\dfrac{d}{dx}\bigl(\arccos x\bigr)

and the main reason for that sign? a) Positive, because both the numerator

1

and the denominator

\sqrt{1-x^{2}}

are positive. b) Negative, because the numerator is

-1

while the denominator

\sqrt{1-x^{2}}

is positive. c) Undefined, because

\sqrt{1-x^{2}}

equals

0

when

x\to 1

, so the derivative has no finite value on

(0,1)

. d) Positive, because the numerator

-1

and the denominator

\sqrt{1-x^{2}}

are both negative, giving a positive quotient.