First slide

Derivatives

Question

$\cos^{- 1} (\frac{y}{b}) = 2 \log (\frac{x}{2}), x > 0 \Rightarrow x^{2} \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} =$

Moderate

A

$4 y$
B

$- 4 y$
C

0
D

$8 y$

Solution

$\begin{matrix} \cos^{- 1} \frac{y}{b} = 2 \log \frac{x}{2} \Rightarrow \frac{- 1}{\sqrt{1 - y^{2} / b^{2}}} \times \frac{1}{b} \frac{d y}{d x} = \frac{2}{x} \Rightarrow \frac{- 1}{\sqrt{b^{2} - y^{2}}} \frac{d y}{d x} = \frac{2}{x} \Rightarrow x \frac{d y}{d x} = - 2 \sqrt{b^{2} - y^{2}} \\ \Rightarrow x^{2} y_{1}^{2} = 4 (b^{2} - y^{2}) \Rightarrow x^{2} 2 y_{1} y_{2} + 2 x y_{1}^{2} = - 8 y y_{1} \Rightarrow x^{2} y^{2} + x y_{1} = - 4 y \end{matrix}$

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