The nth derivative of x^{n} is equal to n!. The nth derivative of x^n is denoted by $\frac{d^n}{dx^n}\left( x^n\right)$, and its formula is given as follows:

$\boxed{\dfrac{d^n}{dx^n}\left( x^n\right)=n!}$

## nth Derivative of x^{n}

**Question:** Find nth Derivative of x^{n}.

**Answer:**

The nth derivative of x to the power n is obtained by repeatedly using the power rule of differentiation: $\frac{d}{dx}$(x^{n}) = nx^{n-1}.

Let us put

y = x^{n}.

By power rule, the first derivative of x^{n} is

y_{1} = nx^{n-1}.

The second derivative y_{2} of y is obtained by differentiating y_{1} with respect to x. So we have that

y_{2} = n(n-1)x^{n-2}.

Similarly, the third and the fourth order derivatives of x^{n} are respectively equal to

y_{3} = n(n-1)(n-2)x^{n-3}.

y_{4} = n(n-1)(n-2)(n-3)x^{n-4}.

By observing the patterns, we see that the nth^{ }derivative of x^{n} is equal to n(n-1)(n-2)(n-3) … {n-(n-1)}x^{n-n} = n(n-1)(n-2)(n-3) …1 x^{0} = n! because x^{0} = 1.

Hence, the nth derivative of x^{n} is equal to n!.

**Also Read:** nth Derivative of 1/x

## Question-Answer

**Question 1:** If y=x^{3}, then find y_{4}.

**Answer:**

From above, we know that the 3rd derivative of x^{3} is equal to 3! = 6. So the fourth derivative is equal to $\frac{d}{dx}$(x^{3}) = $\frac{d}{dx}$(6) = 0 as the derivative of a constant is zero.

So if y=x^{3}, then y_{4} = 0. That is, the fourth order derivative of x^{3} (x cube) is equal to 0.

**Question 1:** If y=x^{10}, then find y_{11}.

**Answer:**

In a similar method as above, if y=x^{10}, then y_{11} = 0.

## FAQs

### Q1: What is nth Derivative of x^{n} (x to power n)?

**Answer:** The nth derivative of x^{n} (x to the power n) is equal to n!.