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Power Rule

Last Updated : 31 Jan, 2024
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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.

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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: 

Power Rule Formula

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 \dfrac{d}{dx}x^n=nx^{n-1}\\

\implies \dfrac{d}{dx}x^{101}=101x^{100}\\\qquad\\ \sqrt{2}

Example: Find the derivative of 15x6.

Answer:

As \dfrac{d}{dx}x^n=nx^{n-1}

\dfrac{d(15x^6)}{dx}=15(6x^{6-1})=90x^5

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:

\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}}

Example: Find the derivative of √x.

Answer:

\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}}

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

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

Inductive Step

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

Let f(x) = xk+1 = x × xk.

Differentiate using the Product Rule, we get:

f'(x) = xk + x × kxk-1 

⇒ f'(x) = xk + kxk 

⇒ f'(x) = (k+1)xk

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

Using Binomial Theorems

Using the definition of derivative we can write

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

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

(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\\

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

\dfrac{d}{dx}x^n\ =\ nx^{n-1}

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 f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0,        we can differentiate it by taking the derivative of each term using the power rule and adding the results.

Example: Find the derivative of \bold{3x^4- 2x^3+5x^2 -7x+1}    .

Answer:

f'(x) = (d/dx)(3x^4) - (d/dx)(2x^3) + (d/dx)(5x^2) - (d/dx)(7x) + (d/dx)(1)

⇒f′(x) = 12x3−6x2+10x−7+0

So the derivative of f(x) is f′(x) = 12x3 − 6x2 + 10x − 7.

Other Power Rules in Calculus

There are various other power rules used in calculus that are used to solve various problems. Some of the various power rules in calculus are,

  • Power Rule Integration
  • Power Rule Exponents
  • Power Rule Logarithms

Now let’s learn about these power rules in detail.

Power Rule Integration

Power rule in integration is helpful for finding the integral of expressions that are given as, xn, where n is a real number and n ≠ -1. The formula for the integration power rule is,

∫xn dx = xn+1/(n + 1) + C

where n ≠ -1. We can understand this rule using the example discussed below,

(i) ∫10x9 dx 

= 10(x9+1)/(9+1) + C 

= 10x10/10 + C 

= x10 + C

(ii) ∫x-3 dx

= x-3+1/(-3+1) + C 

= -x-2/2+ C 

= -1/2x2 + C

Power Rule Exponents

Power rule in exponents is used we have to find the power of the exponents that are given as,

(xm)n = xmn

We can understand this rule using the example discussed below,

  • (x2)4 = x2×4 = x8
  • (2-3)-3 = 2-3×-3 = 29

Power Rule Logarithms

The power rule in the logarithmic is used to solve the power of any logarithmic function such as,

logn(a)b = b.logn(a)

We can understand this rule using the example discussed below,

  • log8x3 = 3log8x

Read More,

Solved Problems of Power Rule

Problem 1: Find the derivative of f(x) = x5.

Solution:

Using the power rule, we have:

f'(x) = 5x(5-1) = 5x4

Problem 2: Find the derivative of \bold{\frac{1}{\sqrt[3]{x}}}        .

Solution:

\text{Let } f(x) = \frac{1}{\sqrt[3]{x}}\\ \Rightarrow f'(x) = \frac{d}{dx}\frac{1}{\sqrt[3]{x}}\ =\ \frac{d}{dx}x^{\frac{-1}{3}}\\ \Rightarrow f'(x) = \frac{-1}{3}x^{\frac{-1}{3}-1}\\ \Rightarrow f'(x) =\ \frac{-1}{3}x^{\frac{-1-3}{3}}\\ \Rightarrow f'(x) =\ \frac{-1}{3}x^{\frac{-4}{3}}\\ \Rightarrow f'(x) =\frac{-1}{3\sqrt[3]{x^{4}}}

Problem 3: Find the derivative of \bold{{\sqrt[5]{x^7}}}    .

Solution:

\text{Let } f(x) = \sqrt[5]{x^7}\\ \Rightarrow f'(x) = \frac{d}{dx}\sqrt[5]{x^7}\ \\ \Rightarrow f'(x) = \frac{d}{dx}x^{\frac{7}{5}}\\ \Rightarrow f'(x) = \frac{7}{5}x^{\frac{7-5}{5}}\\ \Rightarrow f'(x) = \frac{7}{5}x^{\frac{2}{5}}

Problem 4: Find the derivative of h(x) = x-2/3.

Solution: 

Thus, h'(x) = (1/3)(-2)x-2-1 = (-2/3)x-3 

Therefore, the derivative of h(x) = x-2/3  is -2x-3/3.

Problem 5: Find the derivative of k(x) = (5x2 + 3x)4.

Solution: 

Using the chain rule and power rule together we get,

 k'(x) = 4(5x^2 + 3x)^{4-1}\times  \frac{d}{dx}(5x^2 + 3x)         

\Rightarrow k'(x) =  4(5x^2 + 3x)^{4-1}\times (10x + 3)

Therefore, the derivative of k(x) = (5x2 + 3x)4 is  4(5x2 + 3x)3(10x + 3).

FAQs on Power Rule

Q1: What is the Power Rule?

Answer:

Power Rule is an important formula in the calculation of differentiation, where we can differentiate function with any power.

Q2: What is the Formula for the Power Rule?

Answer:

The mathematical formula of power rule can be written as: 

\bold{\frac{d(x^n)}{dx}=nx^{n-1} }

Q3: What is the Difference between the Power Rule and the Chain Rule?

Answer:

The power rule and chain rule are two very different formulas for the calculation of differentiation, where the power rule is used for functions with exponents whereas the chain rule is used to differentiate the composite functions.

Q4: Can Power Rule be used to find the derivative of a function with a negative exponent?

Answer:

Yes, Power rule can be used to find the derivativeof a function with negative expoenet.

Q5: Can Power Rule be applied to functions with non-integer exponents?

Answer:

Yes, the power rule can be applied to the functions with non-integer exponents.



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