Mastering the Power Rule: A Step-by-Step Guide to Finding Derivatives of Functions with Constant Powers

Power Rule for Derivatives

The power rule is a fundamental rule for finding the derivative of a function that contains a variable raised to a constant power

The power rule is a fundamental rule for finding the derivative of a function that contains a variable raised to a constant power. It allows us to find the derivative of a polynomial function quickly and easily.

The power rule 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 is given by:

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

Let’s go through an example to demonstrate how to apply the power rule:

Example 1:
Find the derivative of f(x) = 4x^3.

To find the derivative, we can simply apply the power rule. According to the power rule, we take the exponent (3) of x, multiply it by the coefficient (4), and then decrease the exponent by 1.

f'(x) = 3 * 4x^(3-1) = 12x^2

So, the derivative of f(x) = 4x^3 is f'(x) = 12x^2.

Now, let’s consider some additional cases:

Case 1: Constant Functions
If we have a constant function, such as f(x) = 5, its derivative is always zero. This is because the derivative measures the rate of change, and a constant function does not change, so its derivative is a constant rate of zero.

f(x) = 5
f'(x) = 0

Case 2: Linear Functions
For a linear function f(x) = mx + b, where m and b are constants, the derivative is simply the slope (m) of the line. This is because the derivative represents the instantaneous rate of change, which is constant for a linear function.

f(x) = mx + b
f'(x) = m

Case 3: Higher Power Functions
For functions with higher powers, we can still apply the power rule. Let’s take an example:

Example 2:
Find the derivative of f(x) = 2x^5.

Using the power rule, we multiply the exponent (5) by the coefficient (2) and decrease the exponent by 1.

f'(x) = 5 * 2x^(5-1) = 10x^4

So, the derivative of f(x) = 2x^5 is f'(x) = 10x^4.

In summary, the power rule is a simple and useful tool for finding derivatives of functions that involve a variable raised to a constant power. By following the formula f'(x) = n * x^(n-1), where n is the exponent, we can obtain the derivative efficiently.

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