Exploring the Power Rule for Derivatives | Simplifying Calculus with Exponential Functions

Power Rule for Derivatives

The power rule for derivatives is a fundamental rule in calculus that allows us to find the derivative of a function that is raised to a power

The power rule for derivatives is a fundamental rule in calculus that allows us to find the derivative of a function that is raised to a power. It applies to functions of the form f(x) = x^n, where n is any real number.

Let’s consider the derivative of a function f(x) = x^n, where n is a constant. To find the derivative, we can use the power rule, which states that the derivative of x^n is given by:

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

This rule tells us that when we differentiate a term with the form x^n, the power n becomes the coefficient in the derivative, and the new power is obtained by subtracting 1 from the original power.

Here are a few examples to illustrate the application of the power rule:

1. f(x) = x^3
Applying the power rule, we differentiate the term x^3 to get:
f'(x) = 3x^(3-1) = 3x^2

2. f(x) = 5x^2
In this case, there is a constant (5) multiplied by x^2. When differentiating, we apply the power rule to x^2, and the constant (5) remains unchanged, giving:
f'(x) = 5 * 2x^(2-1) = 10x

3. f(x) = √x
Here, we have a square root function. To differentiate, we can rewrite the function as x^(1/2) and apply the power rule:
f'(x) = (1/2)x^(1/2 – 1) = (1/2)x^(-1/2) = 1/(2√x)

Note that the power rule only applies when the exponent is a constant. If the exponent itself is a function of x, we need to use the chain rule in conjunction with the power rule.

In summary, the power rule for derivatives is a valuable tool for finding the derivative of functions involving powers of x. It simplifies the process by providing a simple formula to differentiate power functions.

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