Power Rule for Derivatives
The power rule is a basic rule in calculus that allows us to find the derivative of a function raised to a power
The power rule is a basic rule in calculus that allows us to find the derivative of a function raised to a power. Specifically, the power rule states that if we have a function f(x) = x^n, where n is a constant, then its derivative is given by:
f'(x) = n * x^(n-1)
Here are a few examples to illustrate how the power rule works:
1. If we have the function f(x) = x^5, then its derivative is f'(x) = 5 * x^(5-1) = 5 * x^4.
2. If we have the function g(x) = x^2, then its derivative is g'(x) = 2 * x^(2-1) = 2 * x.
3. If we have the function h(x) = x^0, which is essentially the constant function 1, then its derivative is h'(x) = 0 * x^(0-1) = 0.
The power rule is particularly useful because it allows us to find the derivative of any polynomial function by applying the rule to each term separately. Additionally, it can be extended to include negative exponents and fractional exponents using rules of exponentiation.
It’s important to note that the power rule only applies when the exponent is a constant. If the exponent is a function of x, then we need to use the chain rule or other derivative rules to find the derivative.
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