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Higher Order Derivatives

If y=x2-1m, t...

Question

If $y = {(x^{2} - 1)}^{m},$ then the value of (2m)th differential coefficient of y is

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(2m)!

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Solution

The correct option is B (2m)!
Expanding binomially, we get
$y = {(x^{2} - 1)}^{m} = m C 0 x^{2 m} + m C 1 x^{2 m - 1} (- 1) + . . . . .$
On differentiating, all the terms, except first, reduces to zero. Therefore,
$\frac{d^{2 m}}{{dx}^{2 m}} {(x^{2} - 1)}^{m} = m C 0 2 m (2 m - 1) (2 m - 2) . . . . . 1 = (2 m)!$

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