First slide

Derivatives

Question

$If y = x \log |x + \sqrt{1 + x^{2}}| - \sqrt{1 + x^{2}} then \frac{d y}{d x} =$

Moderate

A

$\log (x - \sqrt{1 + x^{2}})$
B

$\frac{1}{2} \log (x + \sqrt{1 + x^{2}})$
C

$\sinh^{- 1} x$
D

$\cosh^{- 1} x$

Solution

$\frac{d}{d x} = x \cdot \frac{1}{\sqrt{1 + x^{2}}} + \log (x + \sqrt{1 + x^{2}}) - \frac{x}{\sqrt{1 + x^{2}}} = \log (x + \sqrt{1 + x^{2}}) = S i n h^{- 1} x .$

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