Power Rule in Derivatives

• Last Updated : 24 Nov, 2022

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 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 a power series with a function’s derivatives.

Examples

Find the derivative of

1. x101

2. 15x6

Power Rule (with rewriting the expression)

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:

Find the derivative of

Justifying the Power Rule

Proof:

Using the definition of derivative we can write

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

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

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