First slide

Derivatives

Question

$Find the differential coefficient of {Tan}^{- 1} (\frac{\sqrt{2 a x - x^{2}}}{a - x}) w . r . to x .$

Moderate

A

$\frac{1}{\sqrt{2 a x - x^{2}}}$
B

$\frac{1}{\sqrt{x^{2} - 2 a x}}$
C

$\frac{a}{\sqrt{1 - 2 a x}}$
D

$\frac{x}{\sqrt{2 a x - x^{2}}}$

Solution

$\begin{array}{l} x = 2 a \sin^{2} θ \\ \Rightarrow y = {Tan}^{- 1} (\frac{2 a \sin θ \cos θ}{a \cos 2 θ}) = 2 θ = 2 \sin^{- 1} \sqrt{\frac{x}{2 a}} \\ ∴ \frac{d y}{d x} = 2 \frac{1}{\sqrt{1 - \frac{x}{2 a}}} \times \frac{1}{\sqrt{2 a}} \times \frac{1}{2 \sqrt{x}} = \frac{1}{\sqrt{2 a x - x^{2}}} \end{array}$

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