Understanding the Power Rule: A Simple Yet Powerful Approach to Calculus Derivatives

derivative power rule

The power rule is a fundamental rule in calculus that allows you to find the derivative of a function that is raised to a constant power

The power rule is a fundamental rule in calculus that allows you to find the derivative of a function that is raised to a constant power. Let’s say you have a function f(x) = x^n, where n is a constant. The power rule states that the derivative of this function is given by:

f'(x) = n * x^(n-1)

In other words, to find the derivative of a function raised to a constant power, you simply multiply the constant power by the coefficient in front of the function and then decrease the power by 1.

Let’s go through an example to demonstrate the power rule. Consider the function f(x) = 3x^2. Using the power rule, we can find its derivative as follows:

f'(x) = 2 * 3x^(2-1)
= 6x^1
= 6x

Therefore, the derivative of f(x) = 3x^2 is f'(x) = 6x.

Here’s another example. Take the function g(x) = 5x^4. Applying the power rule, we get:

g'(x) = 4 * 5x^(4-1)
= 20x^3

So, the derivative of g(x) = 5x^4 is g'(x) = 20x^3.

The power rule can also be extended to functions involving more than one term. For example, if we have a function h(x) = 2x^3 + 4x^2, we can find its derivative by applying the power rule to each term separately:

h'(x) = 3 * 2x^(3-1) + 2 * 4x^(2-1)
= 6x^2 + 8x

Therefore, the derivative of h(x) = 2x^3 + 4x^2 is h'(x) = 6x^2 + 8x.

The power rule is a powerful tool that simplifies finding derivatives of functions raised to constant powers. By using this rule, you can quickly find the derivative of functions with various exponents, making it an essential technique in calculus.

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