The Power Rule is one of the most fundamental tools in calculus differentiation. It provides a simple way to find the derivative of polynomial functions, which are functions consisting of terms like powers of variables. The Power Rule states that if you have a function of the form \(x^n\), where \(n\) is any real number, its derivative is \(nx^{n-1}\). This makes it particularly easy to differentiate terms like \(20x^3\) and \(-36x^2\).
- For \(20x^3\), applying the Power Rule means multiplying the coefficient (20) by the exponent (3), and then reducing the exponent by 1, resulting in \(60x^2\).
- For \(-36x^2\), multiply \(-36\) by the exponent (2), and reduce the exponent by 1, yielding \(-72x\).
In essence, the Power Rule is a quick way to determine how the function changes, or its rate of change at any given point. It simplifies the process of finding higher-order derivatives like the second or third, by allowing you to apply the rule repeatedly.