First slide

Derivatives

Question

$.If x + y = \tan^{- 1} y and y^{11} = f (y) y^{1} then f (y) =$

Moderate

A

$\frac{- 2}{y (1 + y^{2})}$
B

$\frac{1}{y (1 + y^{2})}$
C

$\frac{2}{y (1 + y^{2})}$
D

$\frac{3}{y (1 + y^{2})}$

Solution

$\begin{array}{l} x + y = \tan^{- 1} y \\ 1 + y^{1} = \frac{1}{1 + y^{2}} y^{1} \\ y^{''} = \frac{- 2 y}{{(1 + y^{2})}^{2}} y^{'} + \frac{1}{1 + y^{2}} y^{''} \\ y^{''} (1 - \frac{1}{1 + y^{2}}) = \frac{- 2 y}{{(1 + y^{2})}^{2}} y^{'} \\ y^{''} (\frac{y^{2}}{1 + y^{2}}) = \frac{- 2 y}{{(1 + y^{2})}^{2}} y^{'} \\ y^{''} = \frac{- 2}{y (1 + y^{2})} y^{'} \\ f (y) = \frac{- 2}{y (1 + y^{2})} (\sin c e y^{''} = f (y) y^{'}) \end{array}$

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