Mastering the Difference Rule in Calculus: Derivatives of Functions Involving Subtraction

The difference rule

The difference rule is a basic rule in mathematics that allows us to find the derivative of a function that involves a difference between two other functions

The difference rule is a basic rule in mathematics that allows us to find the derivative of a function that involves a difference between two other functions.

In calculus, the derivative represents the rate of change of a function at a certain point. It measures how the function is changing as the input variable (usually denoted as x) changes. The difference rule helps us find the derivative of a function that involves subtraction or a difference between two other functions.

Formally, let’s say we have two functions f(x) and g(x), and we want to find the derivative of h(x) = f(x) – g(x). The difference rule states that the derivative of h(x) is equal to the difference between the derivatives of f(x) and g(x), i.e., h'(x) = f'(x) – g'(x).

To apply the difference rule, we can take the derivative of each function separately using other derivative rules (such as the power rule, product rule, chain rule, etc.) and then subtract the results. This allows us to find the derivative of the overall function involving the difference.

Let’s go through an example to illustrate the application of the difference rule:

Example:
Consider the function h(x) = 3x^2 – 5x + 2. We want to find the derivative of this function.

To find the derivative, we can break down h(x) into f(x) = 3x^2 and g(x) = 5x – 2.

Now, we differentiate each function separately:
f'(x) = 6x (using the power rule: d/dx (x^n) = n*x^(n-1))
g'(x) = 5 (since the derivative of a constant is zero)

Finally, we apply the difference rule:
h'(x) = f'(x) – g'(x)
= 6x – 5

Therefore, the derivative of h(x) = 3x^2 – 5x + 2 is h'(x) = 6x – 5.

The difference rule is a useful tool in calculus to find the derivative when dealing with functions involving subtraction or a difference between two other functions. It helps us break down complex functions and simplify the process of finding their derivatives.

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