What is the derivative of the polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$?

Primary tabs

Lesson Parent: 
Differentiation IV: Differentiation Formulas Everyone Must Know
Problem: 

What is the derivative of the polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$?

Answer: 

$f'(x)=n\,a_nx^{n-1}+(n-1)\,a_{n-1}x^{n-2}+\cdots+a_1$

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

Recall what a polynomial is here.

We know that <Sign in to see all the formulas> and that $a_0$ is differentiable since it is a constant. Furthermore, if $C$ is a constant and $g(x)$ is differentiable then <Sign in to see all the formulas>. Therefore, each <Sign in to see all the formulas> is differentiable.

Since each term in <Sign in to see all the formulas> is differentiable, $f'(x)$ is the sum of the derivatives of <Sign in to see all the formulas>. Therefore, <Sign in to see all the formulas>.