power rule
In calculus, the power rule is a helpful rule for finding the derivative of a function that is in the form of a power function
In calculus, the power rule is a helpful rule for finding the derivative of a function that is in the form of a power function. The power rule states that if we have a function f(x) = x^n, where n is any real number, the derivative f'(x) can be found by multiplying the exponent by the current coefficient and then subtracting 1 from the exponent.
The power rule can be stated mathematically as follows:
If f(x) = x^n, where n is any real number, then f'(x) = n * x^(n-1).
Let’s go through some examples to understand and apply the power rule:
Example 1: Find the derivative of the function f(x) = x^3.
Applying the power rule, we multiply the exponent 3 by the coefficient 1, and then subtract 1 from the exponent:
f'(x) = 3 * x^(3-1) = 3 * x^2.
Therefore, the derivative of f(x) = x^3 is f'(x) = 3x^2.
Example 2: Find the derivative of the function g(x) = 5x^4.
Using the power rule, we multiply the exponent 4 by the coefficient 5, and then subtract 1:
g'(x) = 4 * 5 * x^(4-1) = 20 * x^3.
So, the derivative of g(x) = 5x^4 is g'(x) = 20x^3.
Example 3: Find the derivative of the function h(x) = 2x^(-1).
Applying the power rule, we multiply the exponent -1 by the coefficient 2, and then subtract 1:
h'(x) = -1 * 2 * x^(-1-1) = -2 * x^(-2).
Thus, the derivative of h(x) = 2x^(-1) is h'(x) = -2x^(-2).
The power rule is a fundamental rule in calculus that allows us to find derivatives of power functions quickly. It is important to remember that the power rule applies only to functions of the form f(x) = x^n, where n is any real number.
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