Question

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

• A. $(n-1)!$
• B. $(n+1)!$
• C. $n!$
• D. $n{\left[\right(n+1\left)\right]}^{n-1}$

Answer 1

The correct option is C

$n!$

Explanation for the correct option:

Finding the $n$th derivative:

Let

$$\begin{gathered} y={(x+1)}^n \\ \frac{dy}{dx}=n{(x+1)}^{n-1} \\ \frac{d^{2}y}{d^{2}x}=n(n-1){(x+1)}^{n-2} \\ \frac{d^{3}y}{d^{3}x}=n(n-1)(n-2){(x+1)}^{n-3} \end{gathered}$$

Similarly nth derivative,

$\frac{d^{n}y}{d^{n}x}=n(n-1)(n-2)\dots (n-(n-1\left)\right){(x+1)}^{n-n}$

$=n(n-1)(n-2)\dots \left(1\right){(x+1)}^0$

$=n!$

Therefore, the $n$th derivative of ${(x+1)}^n$ $=n!$

Hence, option(C) is the correct answer.