Power Rule
The power rule is a fundamental rule in calculus that allows us to find the derivative of a function that is in the form of x raised to a power
The power rule is a fundamental rule in calculus that allows us to find the derivative of a function that is in the form of x raised to a power. It states that if we have a function of the form f(x) = x^n, where n is a constant, then the derivative of this function, denoted as f'(x), is given by:
f'(x) = n * x^(n-1)
This means that when differentiating a function in the form of x raised to a power, we can bring the constant down as a coefficient and reduce the power of x by 1.
Let’s work through an example to better understand the power rule:
Example:
Find the derivative of the function f(x) = 5x^3
Solution:
To find the derivative of f(x), we can apply the power rule. According to the power rule, we need to bring down the exponent as a coefficient and reduce the exponent by 1. Applying this to our example, we have:
f'(x) = 3 * 5x^(3-1)
Simplifying, we get:
f'(x) = 15x^2
Therefore, the derivative of f(x) = 5x^3 is f'(x) = 15x^2.
The power rule is very useful when it comes to differentiating polynomials and functions that involve powers of x. It provides a straightforward method for finding derivatives and is often used as a building block for more complex differentiation techniques.
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