Power Rule in Derivatives
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
![1.\ \ x^{\dfrac{-3}{4}}\\\qquad\\ \dfrac{d}{dx}x^{\dfrac{-3}{4}}\ =\ \dfrac{-3}{4}x^{\dfrac{-3}{4}-1}\ =\ \dfrac{-3}{4}x^{\dfrac{-3-4}{4}}\ =\ \dfrac{-3}{4}x^{\dfrac{-7}{4}} \\\qquad\\\qquad\\ 2. \ \ \sqrt{x}\\\qquad\\ \dfrac{d}{dx}\sqrt{x}\ =\ \dfrac{d}{dx}x^{\dfrac{1}{2}}=\dfrac{1}{2}x^{\dfrac{1}{2}-1}\ =\ \dfrac{1}{2}x^{\dfrac{-1}{2}}\ =\ \dfrac{1}{2\sqrt{x}} \\\qquad\\\qquad\\ 3. \ \ \dfrac{1}{\sqrt[3]{x}}\\\qquad\\ \dfrac{d}{dx}\dfrac{1}{\sqrt[3]{x}}\ =\ \dfrac{d}{dx}x^{\dfrac{-1}{3}}\ =\ \dfrac{-1}{3}x^{\dfrac{-1}{3}-1}\ =\ \dfrac{-1}{3}x^{\dfrac{-1-3}{3}}\ =\ \dfrac{-1}{3}x^{\dfrac{-4}{3}}\ =\dfrac{-1}{3\sqrt[3]{x^{4}}} \\\qquad\\\qquad\\ 4. \ \ \sqrt[5]{x^7}\\\qquad\\ \dfrac{d}{dx}\sqrt[5]{x^7}\ =\ \dfrac{d}{dx}x^{\dfrac{7}{5}}\ =\ \dfrac{7}{5}x^{\dfrac{7-5}{5}}\ =\ \dfrac{7}{5}x^{\dfrac{2}{5}} \\\qquad\\\qquad\\](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-78ed3fa3c85538ecb0979bc892e6ad2a_l3.png "Rendered by QuickLaTeX.com")
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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