Power Rule for Non-Integers
From the above equation and example, you now know how to differentiate a variable raised to a power n. The point to be noted is that n can also be fractional and so the variable could have exponents and these exponents are real numbers. For better understanding check the following examples:
Example: Find the derivative of x^{\frac{-3}{4}}
Answer:
[Tex]\text{Let } f(x) = x^{\frac{-3}{4}}\\ \Rightarrow f'(x) = \frac{d}{dx}x^{\frac{-3}{4}}\\ \Rightarrow f'(x) =\frac{-3}{4}x^{\frac{-3}{4}-1}\\ \Rightarrow f'(x) =\frac{-3}{4}x^{\frac{-3-4}{4}}\\ \Rightarrow f'(x) =\frac{-3}{4}x^{\frac{-7}{4}} [/Tex]
Example: Find the derivative of √x.
Answer:
[Tex]\text{Let } f(x) = \sqrt{x}\\ \Rightarrow f'(x) = \frac{d}{dx}\sqrt{x}\ = \frac{d}{dx}x^{\frac{1}{2}}\\ \Rightarrow f'(x) =\frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{\frac{-1}{2}}\\ \Rightarrow f'(x) = \frac{1}{2\sqrt{x}} [/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