Applications of Power Rule
The power rule states that the derivative of x to the power n is equal to n times x to the power n-1. In other words, if we have a polynomial function [Tex]f(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0, [/Tex] we can differentiate it by taking the derivative of each term using the power rule and adding the results.
Example: Find the derivative of [Tex]\bold{3x^4- 2x^3+5x^2 -7x+1} [/Tex].
Answer:
[Tex]f'(x) = (d/dx)(3x^4) – (d/dx)(2x^3) + (d/dx)(5x^2) – (d/dx)(7x) + (d/dx)(1) [/Tex]
⇒f′(x) = 12x3−6x2+10x−7+0
So the derivative of f(x) is f′(x) = 12x3 − 6x2 + 10x − 7.
Power Rule
Power Rule is a fundamental rule in the calculation of derivatives that helps us find the derivatives of functions with exponents. Exponents can take any form, including any function itself. With the help of the Power Rule, we can differentiate polynomial functions, functions with variable exponents, and many more.
It is a very diverse tool in the arsenal of students who want to learn the process of differentiation. This article covers the Power Rule, including its formula and derivation, solved examples, applications in calculus, and various commonly asked curious questions related to the Power Rule.
Table of Content
- Power Rule Formula
- Power Rule for Non-Integers
- Derivation of Power Rule
- Applications of Power Rule
- Other Power Rules in Calculus