Power Rule
The power rule is a fundamental property in calculus that allows us to differentiate functions of the form f(x) = x^n, where n is a constant
The power rule is a fundamental property in calculus that allows us to differentiate functions of the form f(x) = x^n, where n is a constant.
The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).
To understand this rule, let’s consider a few examples:
Example 1:
Let’s find the derivative of f(x) = x^2.
Using the power rule, we know that the derivative of f(x) = x^2 is f'(x) = 2x^(2-1) = 2x.
Therefore, the derivative of f(x) = x^2 is f'(x) = 2x.
Example 2:
Now, let’s find the derivative of f(x) = x^3.
Using the power rule, we apply the same process. The derivative of f(x) = x^3 is f'(x) = 3x^(3-1) = 3x^2.
Therefore, the derivative of f(x) = x^3 is f'(x) = 3x^2.
Example 3:
What if we have a constant multiple in front of the x raised to a power?
Let’s find the derivative of f(x) = 2x^4.
Using the power rule, we find that the derivative of f(x) = 2x^4 is f'(x) = 4(2)x^(4-1) = 8x^3.
Therefore, the derivative of f(x) = 2x^4 is f'(x) = 8x^3.
The power rule is a handy tool in calculus as it allows us to quickly find the derivative of functions involving powers of x. It is based on the observation that when we differentiate a term with x raised to a power, the resulting derivative will have a coefficient equal to the power multiplied by the original coefficient, and the power reduced by 1.
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