Question

Which expression is the derivative of \[\operatorname{cosec}^{-1}(f(x))\] when \[|f(x)|>1\]?

Options

• \[-\frac{1}{1+\{f(x)\}^{2}}\frac{d}{dx}f(x)\]

• \[\frac{1}{|f(x)|\sqrt{\{f(x)\}^{2}-1}}\frac{d}{dx}f(x)\]

• \[-\frac{1}{\sqrt{1-\{f(x)\}^{2}}}\frac{d}{dx}f(x)\]

• \[-\frac{1}{|f(x)|\sqrt{\{f(x)\}^{2}-1}}\frac{d}{dx}f(x)\]

Answer 1

The derivative of \[\operatorname{cosec}^{-1}(f(x))\] is negative. Its denominator is \[|f(x)|\sqrt{\{f(x)\}^{2}-1}\], with \[|f(x)|>1\].