First slide

Derivatives

Question

$If y = \tan^{- 1} \frac{1}{1 + x + x^{2}} + \tan^{- 1} \frac{1}{x^{2} + 3 x + 3} + \tan^{- 1} \frac{1}{x^{2} + 5 x + 7} + \dots . upto n terms then find the value of y^{1} (0)$

Moderate

A

$\frac{- n^{2}}{1 + n^{2}}$
B

$\frac{n^{2}}{1 + n^{2}}$
C

$\frac{1}{1 + n^{2}}$
D

$\frac{- 1}{1 + n^{2}}$

Solution

$y = {Tan}^{- 1} \frac{x + 1 - x}{1 + x (1 + x)} + {Tan}^{- 1} \frac{(x + 2) - (x + 1)}{1 + (x + 1) (x + 2)} + \dots n terms$
$= {Tan}^{- 1} (x + 1) - {Tan}^{- 1} x + {Tan}^{- 1} (x + 2) - {Tan}^{- 1} (x + \underline{1}) + \dots . + + {Tan}^{- 1} (x + n) - {Tan}^{- 1} (x + (n - 1))$
$y = {Tan}^{- 1} (x + n) - {Tan}^{- 1} x$
$\begin{array}{l} y^{'} = \frac{1}{1 + (x + n)^{2}} - \frac{1}{1 + x^{2}} \\ \Rightarrow y^{'} (0) = \frac{1}{1 + n^{2}} - 1 = \frac{1 - 1 - n^{2}}{1 + n^{2}} \\ y^{'} (0) = \frac{- n^{2}}{1 + n^{2}} \end{array}$

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