Part 1/5:
Understanding the General Power Rule in Calculus
In calculus, the power rule is a fundamental concept used to find the derivative of functions expressed in power form. The specific case of the power rule for positive integers has been previously explored, but in this article, we will expand our understanding to cover all real numbers using a method called logarithmic differentiation.
The General Power Rule
The general power rule states that if you have a function (y = x^n), where (n) can be any real number, the derivative (y') can be found efficiently.
To derive this rule, we start by taking the natural logarithm of both sides of the equation:
[
\ln(y) = \ln(x^n)
]
To handle the exponent effectively, we apply logarithmic properties, resulting in:
[