Question

find nth Derivative of y = 4x÷
(x-1)^2(x+1)^2​

Answer 1

$\tt : \implies \: Let \: y \: = \: \dfrac{4x}{ {(x + 1)}^{2} {(x - 1)}^{2} }$

$\tt : \implies \: y \: = \dfrac{ {(x + 1)}^{2} - {(x - 1)}^{2} }{ {(x + 1)}^{2} {(x - 1)}^{2} }$

$\tt : \implies \: y = \dfrac{1}{ {(x + 1)}^{2} } - \dfrac{1}{(x - 1)^{2} }$

Now, Differentiate w. r. t. x, we get

$\tt : \implies \: y_1 = \dfrac{( - 2)}{ {(x - 1)}^{3} } - \dfrac{( - 2)}{ {(x + 1)}^{3} }$

Now, again differentiating both sides w. r. t. x, we get

$\tt : \implies \: y_2 = \dfrac{( - 2)( - 3)}{ {(x - 1)}^{4} } - \dfrac{( - 2)( - 3)}{ {(x + 1)}^{4} }$

$\tt : \implies \: y_2 = \dfrac{ {( - 1)}^{2} 3!}{ {(x - 1)}^{4} } - \dfrac{ {( - 1)}^{2}3! }{ {(x + 1)}^{4} }$

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Now, again differentiating till nth derivative we get

$\tt : \implies \: y_n = \dfrac{ {( - 1)}^{n} (n + 1)!}{ {(x - 1)}^{(n + 2)} } - \dfrac{ {( - 1)}^{n}(n + 1)! }{ {(x + 1)}^{n + 2} }$