Power Rule Formula
The power rule is a commonly used rule in derivatives. The power rule basically states that the derivative of a variable raised to a power n is n times the variable raised to power n-1. The mathematical formula of the power rule can be written as:
Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates to a power series with a function’s derivatives.
Example: Find the derivative of x101.
Answer:
As [Tex]\dfrac{d}{dx}x^n=nx^{n-1}\\ [/Tex]
[Tex]\implies \dfrac{d}{dx}x^{101}=101x^{100}\\\qquad\\ \sqrt{2}[/Tex]
Example: Find the derivative of 15x6.
Answer:
As [Tex]\dfrac{d}{dx}x^n=nx^{n-1} [/Tex]
[Tex]\dfrac{d(15x^6)}{dx}=15(6x^{6-1})=90x^5 [/Tex]
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