Power Rule
The power rule is a fundamental rule in calculus used to differentiate functions of the form f(x) = x^n, where n is a constant
The power rule is a fundamental rule in calculus used to differentiate functions of the form f(x) = x^n, where n is a constant. It states that if f(x) = x^n, then the derivative of f(x) with respect to x is given by f'(x) = nx^(n-1).
To understand how the power rule works, let’s consider a few examples:
1. If f(x) = x^2, using the power rule, we take the exponent 2 and multiply it by x^(2-1) = x^1. So, the derivative of f(x) is f'(x) = 2x.
2. If g(x) = x^3, we apply the power rule similarly. The derivative of g(x) is g'(x) = 3x^(3-1) = 3x^2.
3. For h(x) = x^4, the power rule gives h'(x) = 4x^(4-1) = 4x^3.
It is important to note that the power rule only applies when the exponent is a constant. If the exponent is a variable or a function, more advanced techniques like the chain rule or product rule might be required.
The power rule is a powerful tool because it allows us to quickly find the derivative of functions involving powers of x. It is used extensively in calculus to solve various problems involving rates of change and optimization. Understanding and applying the power rule is crucial in mastering calculus and its related applications.
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