Question

Which derivative and condition are correct for \[\cos^{-1}x\]?

विकल्प

• \[\frac{d}{dx}(\cos^{-1}x)=\frac{1}{\sqrt{1-x^2}},\quad |x|<1\]

• \[\frac{d}{dx}(\cos^{-1}x)=-\frac{1}{\sqrt{1-x^2}},\quad |x|<1\]

• \[\frac{d}{dx}(\cos^{-1}x)=-\frac{1}{1+x^2},\quad x\in\mathbb{R}\]

• \[\frac{d}{dx}(\cos^{-1}x)=-\frac{1}{|x|\sqrt{x^2-1}},\quad |x|>1\]

Answer 1

For \[\cos^{-1}x\], the denominator is \[\sqrt{1-x^2}\]. A negative sign is especially important, and the condition is \[|x|<1\].