power rule
The power rule is a fundamental rule in calculus that allows you to differentiate functions involving powers of a variable
The power rule is a fundamental rule in calculus that allows you to differentiate functions involving powers of a variable. It states that if you have a function of the form f(x) = x^n, where n is a constant, then the derivative of f(x) is given by f'(x) = nx^(n-1).
To understand the power rule, let’s look at a few examples:
Example 1: Differentiate the function f(x) = x^4.
Solution: According to the power rule, the derivative of f(x) = x^4 is f'(x) = 4x^(4-1) = 4x^3.
Example 2: Differentiate the function f(x) = x^2.
Solution: Using the power rule, the derivative of f(x) = x^2 is f'(x) = 2x^(2-1) = 2x.
Example 3: Differentiate the function f(x) = x^(-3).
Solution: Applying the power rule, the derivative of f(x) = x^(-3) is f'(x) = -3x^(-3-1) = -3x^(-4) = -3/x^4.
It’s important to note that the power rule only applies when the variable is raised to a constant power. If the exponent depends on the variable itself, more advanced methods like the chain rule or logarithmic differentiation may be required.
The power rule is a powerful tool in calculus that enables us to differentiate polynomial functions quickly and easily. By applying this rule, you can find the derivative of any function involving a power of a variable.
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