The nth derivative of xn 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 xn
Question: Find nth Derivative of xn.
Answer:
The nth derivative of x to the power n is obtained by repeatedly using the power rule of differentiation: $\frac{d}{dx}$(xn) = nxn-1.
Let us put
y = xn.
By power rule, the first derivative of xn is
y1 = nxn-1.
The second derivative y2 of y is obtained by differentiating y1 with respect to x. So we have that
y2 = n(n-1)xn-2.
Similarly, the third and the fourth order derivatives of xn are respectively equal to
y3 = n(n-1)(n-2)xn-3.
y4 = n(n-1)(n-2)(n-3)xn-4.
By observing the patterns, we see that the nth derivative of xn is equal to n(n-1)(n-2)(n-3) … {n-(n-1)}xn-n = n(n-1)(n-2)(n-3) …1 x0 = n! because x0 = 1.
Hence, the nth derivative of xn is equal to n!.
Also Read: nth Derivative of 1/x
Question-Answer
Question 1: If y=x3, then find y4.
Answer:
From above, we know that the 3rd derivative of x3 is equal to 3! = 6. So the fourth derivative is equal to $\frac{d}{dx}$(x3) = $\frac{d}{dx}$(6) = 0 as the derivative of a constant is zero.
So if y=x3, then y4 = 0. That is, the fourth order derivative of x3 (x cube) is equal to 0.
Question 1: If y=x10, then find y11.
Answer:
In a similar method as above, if y=x10, then y11 = 0.
FAQs
Q1: What is nth Derivative of xn (x to power n)?
Answer: The nth derivative of xn (x to the power n) is equal to n!.