Differentiation: Understanding and Applying the Difference Rule in Math

The difference rule

The difference rule is a mathematical rule used to differentiate a function that is formed by subtracting one function from another

The difference rule is a mathematical rule used to differentiate a function that is formed by subtracting one function from another. This rule is also known as the subtraction rule or the rule of differences.

In order to understand the difference rule, it is important to have a basic understanding of differentiation. Differentiation is the process of finding the derivative of a function, which gives us the rate of change of the function at any given point.

The general form of the difference rule is as follows:

d/dx [f(x) – g(x)] = f'(x) – g'(x)

Here, f(x) and g(x) are two differentiable functions, and f'(x) and g'(x) are their respective derivatives with respect to x.

To apply the difference rule, you follow these steps:

1. Differentiate f(x) and g(x) separately.
2. Subtract the derivative of g(x) from the derivative of f(x).

Let’s look at an example to better understand how the difference rule works:

Example: Find the derivative of the function h(x) = 3x^2 – 2x.

Solution:
Using the difference rule, we differentiate the two terms separately:
– The derivative of 3x^2 is obtained by applying the power rule of differentiation, which gives us 6x.
– The derivative of 2x is simply 2.

Now, we subtract the derivative of 2x from the derivative of 3x^2:
h'(x) = 6x – 2.

Therefore, the derivative of h(x) is 6x – 2.

The difference rule can be very useful when dealing with functions that involve subtraction. It allows us to differentiate these functions efficiently by differentiating the individual terms and subtracting their derivatives.

It’s important to note that the difference rule only applies to subtraction, and not to addition. For functions involving addition, the sum rule is used instead.

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