$n$ th derivative of ${(x + 1)}^{n}$ is equal to

(n - 1)!

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(n + 1)!

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n!

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n {[(n + 1)]}^{n - 1}

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Solution

The correct option is C $n!$
Let $y = {(x + 1)}^{n}$
$∴ \frac{d y}{d x} = n {(x + 1)}^{n - 1}$
$\frac{d^{2} y}{d x^{2}} = n (n - 1) {(x + 1)}^{n - 2}$
Similarly,
$\frac{d^{n} y}{d x^{n}} = n (n - 1) (n - 2) \dots 3 \cdot 2 \cdot 1 \times (x + 1)^{(n - n)}$
$= n!$

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