Differentiate In $t a n x$ with respect to ${sin}^{- 1} (e^{x})$

\frac{e^{- x} \sqrt{1 - e^{2 x}}}{sin x \cdot cos x}

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\frac{e^{- x} \sqrt{1 - e^{2 x}}}{sin x \cdot cot x}

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\frac{e^{- x} \sqrt{1 + e^{x}}}{sin x \cdot cos x}

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\frac{e^{x} \sqrt{1 + e^{2 x}}}{sin x \cdot sec x}

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Solution

The correct option is B $\frac{e^{- x} \sqrt{1 - e^{2 x}}}{sin x \cdot cos x}$
$\frac{d (l n t a n x)}{d {sin}^{- 1} e^{x}} = \frac{d (l n (t a n x))}{d x} \div \frac{d {sin}^{- 1} e^{x}}{d x}$
$= \frac{\frac{1. {sec}^{2} x}{t a n x}}{\frac{e^{x}}{\sqrt{{1 - e}^{2 x}}}}$
$= \frac{1}{s i n x c o s x} e^{- x} \sqrt{{1 - e}^{2 x}}$ [A]

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