For , which of the following correctly describes the sign of and the main reason for…

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

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.