First slide

Differentiability

Question

$If \sec (\frac{x - y}{x + y}) = a then \frac{d y}{d x} =$

Moderate

A

$\frac{y}{x}$
B

$- \frac{y}{x}$
C

$\frac{x}{y}$
D

$- \frac{x}{y}$

Solution

$\begin{array}{l} \frac{x - y}{x + y} = \sec^{- 1} a \\ \frac{x - y + x + y}{x - y - x - y} = \frac{\sec^{- 1} a + 1}{\sec^{- 1} a - 1} \\ \frac{- 2 x}{2 y} = \frac{\sec^{- 1} a + 1}{\sec^{- 1} a - 1} \\ \frac{y}{x} = \frac{1 - \sec^{- 1} a}{1 + \sec^{- 1} a} \\ \frac{x \frac{d y}{d x} - y}{x^{2}} = 0 \\ \Rightarrow \frac{d y}{d x} = \frac{y}{x} \end{array}$

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