Power Rule

Power rule in calculus is a method of differentiation that is used when an algebraic expression with power needs to be differentiated. In simple words, we can say that the power rule is used to differentiate algebraic expressions of the form xⁿ, where n is a real number. To differentiate xⁿ, we simply multiply the power n by the expression and reduce the power by 1. So, the general power rule derivative formula is given by, d(xⁿ)/dx = nx^n-1.

Further in this article, we will explore the concept of power rule derivatives and its formula. We will also prove the general formula of the power rule and understand its application with the help of various solved examples for a better understanding. Also, we will discuss some other power rules in calculus used in integration, exponents, and logarithmic functions.

The power rule for differentiation is used to differentiate algebraic expressions with power, that is if the algebraic expression is of form xⁿ, where n is a real number, then we use the power rule to differentiate it. Using this rule, the derivative of xⁿ is written as the power multiplied by the expression and we reduce the power by 1. So, the derivative of xⁿ is written as nx^n-1. This implies the power rule derivative is also used for fractional powers and negative powers along with positive powers.

The power rule can be used to differentiate polynomials. Power rule in calculus is used for various types of functions and for different purposes which we will discuss later in this article.

We can write the general power rule formula as the derivative of x to the power n is given by n multiplied by x to the power n minus 1. Mathematically, the general formula for the differentiation of algebraic expressions using the of power rule is given by, d(xⁿ)/dx OR (xⁿ)' = nx^n-1, where n is a real number.

Now we know the formula for the power rule derivative, let us now prove the formula using different methods. We will prove the general formula for the power rule using the principle of mathematical induction and the binomial theorem.

Power Rule Proof Using Mathematical Induction

Using the principle of mathematical induction, we will prove the formula d(xⁿ)/dx = nx^n-1 for positive integral values of n. Here our statement is P(n): d(xⁿ)/dx = nx^n-1. First, we will prove this for n = 1. Then, assume P(n) to be true for n = k, we will prove it for n = k+1.

Step 1: Assume n = 1, then we have LHS = dx/dx = 1 (Because derivative of x is equal to 1). Also, RHS = 1.x^1-1 = 1.x⁰ = 1. Therefore, we have LHS = RHS. Hence, P(1) is true. --- (1)

Step 2: Assume P(k) is true, that is, d(x^k)/dx = kx^k-1 --- (2)

Step 3: Now, we will prove P(n) is true for n = k + 1, that is, we need to prove d(x^k+1)/dx = (k+1)x^k

Consider LHS = d(x^k+1)/dx

= d(x^k.x)/dx --- [Using Exponent Rules]

= d(x^k)/dx × x + dx/dx × x^k --- [Using Product Rule]

= kx^k-1 × x + 1 × x^k [From (1) and (2)]

= kx^k-1+1 + x^k

= kx^k + x^k

= (k+1) x^k

= RHS

Hence, P(k+1) is true.

Therefore, using the principle of mathematical induction, we have proved that P(n): d(xⁿ)/dx = nx^n-1 is true for all natural numbers, and hence, we have proved the power rule formula for differentiation.

Proof of Power Rule Using Binomial Theorem

In this section, we will prove the general power rule formula for differentiation using the binomial theorem formula. The formula for binomial theorem is given by, (x + y)ⁿ = ⁿC₀ xⁿ + ⁿC₁ x^n-1 y + ⁿC₂ x^n-2 y² + ⁿC₃ x^n-3 y³ + ⁿC₄ x^n-4 y⁴ + ... + ⁿC_n yⁿ. We will use the first principle of differentiation to prove the formula and hence, use the binomial formula to arrive at the result. According to the first principle, the derivative of f(x) = xⁿ is given by,

f'(x) = lim_h→0 [(x + h)ⁿ - xⁿ] / h

= lim_h→0 [(ⁿC₀ xⁿ + ⁿC₁ x^n-1 h + ⁿC₂ x^n-2 h² + ⁿC₃ x^n-3 h³ + ⁿC₄ x^n-4 h⁴ + ... + ⁿC_n hⁿ) - xⁿ] / h

= lim_h→0 [ⁿC₁ x^n-1 h + ⁿC₂ x^n-2 h² + ⁿC₃ x^n-3 h³ + ⁿC₄ x^n-4 h⁴ + ... + ⁿC_n hⁿ] / h

= lim_h→0 [ⁿC₁ x^n-1 h + ⁿC₂ x^n-2 h² + ⁿC₃ x^n-3 h³ + ⁿC₄ x^n-4 h⁴ + ... + ⁿC_n hⁿ] / h

= lim_h→0 [ⁿC₁ x^n-1 + ⁿC₂ x^n-2 h + ⁿC₃ x^n-3 h² + ⁿC₄ x^n-4 h³ + ... + ⁿC_n h^n-1] / h

= nx^n-1

Hence, we have proved the power rule formula for differentiation for positive integers n.

Power Rule Formula Proof for Negative Integers

Next, we will generalize the power rule formula for negative integers. Assume n = -m, where m is a natural number. Then, n is a negative integer. So, the derivative of xⁿ is given by,

d(xⁿ)/dx = d(x^-m)/dx

= d(1/x^m)/dx --- [Using exponent rules]

= [d(1)/dx × x^m - d(x^m)/dx] / (x^m)² --- [Using Quotient Rule of Derivatives]

= [0 - mx^m-1] / (x^m)²

= - m x^m-1 / x^2m --- [Using Exponent Rule: (a^m)ⁿ = a^mn]

= -mx^m-1-2m

= -mx^-m-1

= nx^{n - 1} --- [Because n = -m]

Hence, for all negative integers, the power rule formula d(xⁿ)/dx = nx^n-1 holds true. Similarly, we can use the implicit differentiation method to prove the formula for rational exponents as well. Hence, we can conclude that the formula d(xⁿ)/dx = nx^n-1 is true for all real numbers n.

Now, in this section, we understand how to apply the power rule for the differentiation of algebraic expressions (or polynomials) including terms of the form xⁿ. Let us consider an example for the same. Consider f(x) = 3x⁴ - 2x^-2 + x - 8. We will find the derivative of f(x) = 3x⁴ - 2x^-2 + x - 8 using the power rule formula and in this expression the power of x is both positive and negative. To find the derivative of f(x) = 3x⁴ - 2x^-2 + x - 8, we will first use the fact that the derivative of sum of functions is equal to the sum of the derivatives of the functions, that is, (u + v)' = u' + v'. Therefore, we have

f'(x) = d(3x⁴ - 2x^-2 + x - 8)/dx

= d(3x⁴)/dx - d(2x^-2)/dx + dx/dx - d(8)/dx

= 3d(x⁴)/dx - 2d(x^-2)/dx + dx/dx - d(8x⁰)/dx --- [Because d(af(x))/dx = ad(f(x))/dx; we can write 1 as x⁰]

= 3 × 4x^4-1 - 2 × (-2)x^-2-1 + 1x^1-1 - 0x^0-1 --- [Applying the Power Rule Derivative Formula]

= 12x³ + 4x^-3 + 1x⁰ - 0

= 12x³ + 4x^-3 + 1

Hence, we have determined the derivative of a polynomial including terms of the form xⁿ using the power rule. Let us now consider an algebraic expression with rational exponents and apply the formula to find its derivative. Consider the polynomial g(x) = x^-4 + x^3/4 -7x^1/9 + 3. Now, we will find the derivative of g(x) using the power rule.

g'(x) = d(x^-4 + x^3/4 -7x^1/9 + 3)/dx

= d(x^-4)/dx + d(x^3/4)/dx - 7d(x^1/9)/dx + d(3x⁰)/dx

= -4x^-4-1 + (3/4) x^{3/4 - 1} - 7 × (1/9) x^1/9-1 + 3 × 0x^0-1

= -4x^-5 + (3/4) x^-1/4 - (7/9) x^-8/9 + 0

= -4x^-5 + (3/4) x^-1/4 - (7/9) x^-8/9

Hence, the derivative of g(x) = x^-4 + x^3/4 - 7x^1/9 + 3 using the power rule is -4x^-5 + (3/4) x^-1/4 - (7/9) x^-8/9.

We study different power rules in calculus which are used in differentiation, integration, for simplifying exponents and logarithmic functions. Let us discuss them in brief below to understand their formula and application in calculus.

Power Rule Integration

The power rule in integration is used to find the integral of expressions of the form xⁿ, where n is a real number and n ≠ -1. The formula for integration power rule is given by, ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, where n ≠ -1. Let us consider a few examples of this formula to understand this rule better.

• ∫x⁷ dx = x⁷⁺¹/(7+1) + C = x⁸/8 + C
  • ∫x^-2 dx = x^-2+1/(-2+1) + C = -x^-1+ C = -1/x + C

Power Rule For Exponents

The power rule for exponents is used when an exponent is raised to a power. For a positive integer x and integers m and n, we have the formula (x^m)ⁿ = x^mn. In this formula, we have two powers m and n which are multiplied by the simplification of the expression. Let us consider some examples to understand it better:

• (x²)³ = x^2×3 = x⁶
• (2^-2)^-1 = 2^-2×-1 = 2² = 4

Power Rule For Logarithms

The power rule for logarithms is used to simplify the logarithmic expressions with powers. When the argument of a logarithm has an exponent in it, that exponent can be brought to in front of the logarithm. Mathematically, we can write the formula for the power rule for logarithms as log_m(a^b) = b log_m(a). For example, we can write:

• log₃x² = 2 log₃x
• log₂5³ = 3 log₂5

Important Notes on Power Rule:

• The general formula for power rule derivative is d(xⁿ)/dx = nx^n-1
• Power rule is used to differentiate algebraic expressions for the form xⁿ and, hence is used to differentiate polynomials.
• n can be any real number value whenever the power rule formula is applied to find the derivative of expressions of the form xⁿ.
• We can derive the following things using the power rule derivatives:
  The derivative of x is 1.
  The derivative of any constant is 0.

☛ Related Topics:

• Differentiation and Integration
• Derivative Rules
• Derivative Calculator

What is Power Rule Derivatives?

Power rule of derivatives is a method of differentiation that is used when a mathematical expression with an exponent needs to be differentiated. It is used when we are given an expression of the form xⁿ and its derivative is to be determined. It says, d/dx(xⁿ) = nx^n-1.

What is the General Formula for Power Rule Derivative?

The formula for the power rule derivative is d(xⁿ)/dx = nx^n-1, where n is a real number. This formula helps to find the derivative of expressions of the form xⁿ and hence, can be used to find derivatives of polynomials including such terms.

How Do You Derive the Power Rule?

We can derive the power rule for derivatives using the principle of mathematical induction and binomial theorem along with the first principle of derivatives. We can also generalize the power rule formula for rational exponents and negative integers by using the formula for positive integers.

How to Use the Power Rule For Derivatives?

We can use the power rule for derivatives by applying the formula d(xⁿ)/dx = nx^n-1 to the algebraic expressions of the form xⁿ. Since differentiation is a linear operation, we can use this formula to find the derivative of polynomials as well.

How to Find the Derivative of a Fraction Using the Power Rule?

To find the derivative of a fraction using the power rule, we can simplify or rationalize the fraction and express in terms of xⁿ to find its derivative using the power rule. For example, we can simplify the expression 3/x² and write it as 3x^-2 to find its derivative. Using power rule, we have d(3/x²)/dx = 3d(x^-2)/dx = 3 (-2) x^-2-1 = -6x^-3.

What is the Power Rule for Exponents?

The power rule for exponents is used when an exponent is raised to a power. For a positive integer x and integers m and n, we have the formula (x^m)ⁿ = x^mn.

How to Integrate Using Power Rule?

We can integrate algebraic expressions of the form xⁿ using the "power rule of integration" formula ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where C is the integration constant.

What is Zero Power Rule?

Zero power rule states that the value of an expression with any base with the power equal to zero is equal to one. That is, we have x⁰ = 1, for real numbers x except for x = 0.

How to Use Power Rule With Square Roots?

To apply the power rule with square roots, we simply replace the square root with 1/2 in the formula d(xⁿ)/dx = nx^n-1. For f(x) = √x, the derivative of f(x) using the power rule is f'(x) = d(√x)/dx = (1/2) x^{1/2 - 1} = (1/2) x^-1/2.