Question

find the nth derivative

Answer 1

The nth Derivative of a Function

Let f:[a,b]→R be differentiable. Then f′(x) exists and is itself a function. It is very possible that f′ is also differentiable and we can differentiate f′ to get (f′)′=f′′.

Definition: Let f:[a,b]→R be differentiable. If f′ is also differentiable then f is said to be 2-times Differentiable its derivative is called the Second Derivative of f and is denoted f′′=(f′)′.

We can define an n-times differentiable function and the nth derivative of f, denoted f(n), similarly, provided that the previous derivatives exist.

For example, if f(x)=x3 then f′(x)=3x2 and f′′(x)=[f′(x)]′=[3x2]′=6x.

Theorem 1: If f is a polynomial of degree n then f(n) is a constant function and f(n+1) is the zero function.

   Proof: Let f be a polynomial of degree n. Then f has the form:

(1)

f(x)=a0+a1x+...+anxn

   If we differentiate f we get:

(2)

f′(x)=a1+2a2x+...+nanxn−1

   So differentiating a polynomial reduces the degree of all of its terms by 1. So f(n) will be a constant. Namely:

(3)

f(n)(x)=n!an

   And furthermore, f(n+1)(x)=0 since the derivative of a constant is equal to 0.