Derivation of Power Rule

We can derive the formula for the power rule using two methods, which are as follows:

• Using the Principle of Mathematical Induction
• Using the Binomial Theorem

Using the Principle of Mathematical Induction

The Power Rule states that if f(x) = xⁿ, where n is a positive integer, then f'(x) = nx^n-1.

Base Case

Let n=1. Then f(x) = x.

and f'(x) = 1, which is equal to the derivative of x.

Thus, the base case is true.

Inductive Hypothesis

Let us assume that the Power Rule holds true for n=k, where k is an arbitrary positive integer. 

Therefore, if f(x) = x^k, then f'(x) = kx^k-1.

Inductive Step

We need to show that the Power Rule holds for n=k+1. 

Let f(x) = x^k+1 = x × x^k.

Differentiate using the Product Rule, we get:

f'(x) = x^k + x × kx^k-1 

⇒ f'(x) = x^k + kx^k 

⇒ f'(x) = (k+1)x^k

Thus by induction, the power rule holds true for all natural numbers.

Using Binomial Theorems

Using the definition of derivative we can write

[Tex]\dfrac{d}{dx}x^n\ as\ \lim\limits_{x\rarr0}\dfrac{(x+\triangle x)^n-x^n}{\triangle x}\\\qquad\\   [/Tex]

By using the binomial theorem we expand (x + △x)ⁿ th term

[Tex](x+\triangle x)^n\ term\\\qquad\\ \lim\limits_{x\rarr0}\dfrac{(x+\triangle x)^n-x^n}{\triangle x}\\\qquad\\ =\ \lim\limits_{x\rarr0}\dfrac{(\dbinom{n}{0}x^n+\dbinom{n}{1}x^{n-1}\triangle x+\dbinom{n}{2}x^{n-2}\triangle x^2….+\dbinom{n}{n}\triangle x^n)-x^n}{\triangle x}\\\qquad\\ =\ \lim\limits_{x\rarr0}\dfrac{\dbinom{n}{1}x^{n-1}\triangle x+\dbinom{n}{2}x^{n-2}\triangle x^2….+\dbinom{n}{n}\triangle x^n}{\triangle x}\\\qquad\\ = \ \lim\limits_{x\rarr0}\dbinom{n}{0}x^n+\dbinom{n}{1}x^{n-1}+\dbinom{n}{2}x^{n-2}\triangle x….+\dbinom{n}{n}\triangle x^n-1)-x^n\\\qquad\\ = \ \binom{n}{1}x^{n-1}\ =\ nx^{n-1} \\\qquad\\     [/Tex]

Only the first term remained as it does not contain an △ x term hence,

[Tex]\dfrac{d}{dx}x^n\ =\ nx^{n-1}  [/Tex]

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.