Mastering the Difference Rule: Calculating Derivatives with the Power of Subtraction

Difference Rule

The difference rule is a basic rule in mathematics used to differentiate the difference between two functions

The difference rule is a basic rule in mathematics used to differentiate the difference between two functions. It allows us to find the derivative of a function that is obtained by taking the difference of two other functions.

The difference rule states that if we have two differentiable functions, let’s call them f(x) and g(x), then the derivative of their difference, denoted as [f(x) – g(x)], is equal to the difference of their individual derivatives, i.e., f'(x) – g'(x).

In mathematical notation, if we have [f(x) – g(x)], the derivative with respect to x is given by:

d/dx [f(x) – g(x)] = d/dx [f(x)] – d/dx [g(x)] = f'(x) – g'(x)

To apply the difference rule, you need to know the individual derivatives of the functions f(x) and g(x) involved. Once you find these derivatives, you simply subtract them to find the derivative of the function [f(x) – g(x)].

Here’s an example to illustrate the difference rule:

Let’s say we have two functions: f(x) = 3x^2 and g(x) = x^3. We want to find the derivative of their difference.

First, we find the derivatives of f(x) and g(x) separately:
f'(x) = d/dx [3x^2] = 6x
g'(x) = d/dx [x^3] = 3x^2

Using the difference rule, we subtract these derivatives:
[f(x) – g(x)]’ = f'(x) – g'(x) = 6x – 3x^2

So, the derivative of the difference between f(x) and g(x) is 6x – 3x^2.

The difference rule is very useful when we have functions that involve subtraction, allowing us to easily find their derivatives. It is one of the fundamental rules of calculus and is utilized in various applications involving rates of change and optimization problems.

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