Understanding and Applying the Power Rule for Derivatives: A Step-by-Step Guide

Power Rule for Derivatives

The power rule for derivatives is a formula used to find the derivative of a function that is in the form of a power function

The power rule for derivatives is a formula used to find the derivative of a function that is in the form of a power function. It states that for any real number n and a differentiable function f(x), the derivative of f(x) raised to the power of n is equal to n times f(x) raised to the power of n-1, multiplied by the derivative of f(x). Mathematically, it can be expressed as:

d/dx [f(x)^n] = n*f(x)^(n-1) * f'(x)

To understand and apply the power rule, let’s go through a step-by-step example:

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

Step 1: Identify the function and the power.
In our example, the function is f(x) = 3x^4. The power is 4.

Step 2: Apply the power rule.
Using the power rule, we differentiate the function by multiplying the power by the coefficient and reducing the power by 1. We also multiply it by the derivative of the base function, in this case, x.

The derivative of f(x) = 3x^4 can be calculated as follows:
d/dx [3x^4] = 4*3*x^(4-1) * d/dx [x]

Simplifying further, we have:
d/dx [3x^4] = 12x^3 * d/dx [x]

Step 3: Simplify the expression.
Since the derivative of the base function x is equal to 1, we can simplify the expression:
d/dx [3x^4] = 12x^3 * 1

Result:
The derivative of f(x) = 3x^4 is given by d/dx [3x^4] = 12x^3.

Therefore, the slope of the function at any x value can be determined using this derivative.

It is important to note that the power rule only applies when the exponent is a constant. If the exponent is a variable or a function of x, then the power rule cannot be used directly and other differentiation techniques need to be applied.

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