The Difference Rule is the opposite of the Sum Rule. In business calculus, we apply the Difference Rule when we are subtracting two or more functions.

You know you need to use the Difference Rule when you see some sort of subtraction of functions happening.

Here are some examples of subtracting functions:

$$f(x)=204x^{10}-5x^4$$

$$f(x)=10x^4-20x-104,000$$

$$f(x)=x^3-x^{(1/2)}-3x-5$$

In terms of application, the Difference Rule is very similar to the Sum Rule, except that we are dealing with minus signs instead of plus signs. Learn how to apply it below.

In words: The derivative of a difference (subtraction) of functions is equal to the difference of the individual derivatives

In math notation: If $f(x)=g(x)-...-h(x)$ then $f'(x)=g'(x)-...-h'(x)$

Let's examine a sample problem below.

Given that $f(x)=6x^{10}-5x^3$,

Question 1Find $f'(x)$

The function $f(x)$ is the difference of two functions, which are $6x^{10}$ and $5x^3$. These two functions are subtracting, as evidenced by the minus sign between them. To find the derivative of this function, we use the Difference Rule. To do so, we get the derivative of both functions and subtract those derivatives together:

$$f'(x)=(6x^{10})'-(5x^3)'$$

We apply the Power Rule to each derivative,

$$f'(x)=10 \cdot 6x^{(10-1)}-3 \cdot 5x^{(3-1)}$$

As usual, we simplify to get our final answer,

$$f'(x)=60x^9-15x^2$$

Sometimes, there will be functions that have both sums and differences put together. For example,

$$f(x)=5x^4+20x^3-11x^{-2}-2x+$$

$$5$$

In this function, the first two terms are adding, but the second and third are subtracting, and so on. How do we tackle this? We need to apply both the Sum Rule and the Difference Rule. An easy way to remember this is to just get each derivative, and keep the existing sign in front of each term. We first get the derivative of each,

$$f'(x)=(5x^4)'+(20x^3)'-(11x^{-2})'$$

$$-(2x)'+(5)'$$

If you look at each term separately, you will see that some are positive (like $5x^4$), and some are negative like $-11x^{-2}$. Once we get the derivatives, we will just use the same sign for each derivative as the original term had. Below, the blue terms are positive and the red ones are negative, and that gives us the final answer,

$$f'(x)=\color{blue}{20x^3}+\color{blue}{60x^2}+$$

$$\color{blue}{22x^{-3}}-\color{red}2$$

By the way, notice that the derivative $-(11x^{-2})'$ now became positive. Why is that? It's because the original term $-11x^{-2}$ was negative (due to the sign in front of it). However, because the exponent ${-2}$ came down multiplying as per the Power Rule, the derivative became $- (-2) \cdot (11x^{(-2-1)})$, and a negative times a negative is a positive, giving us $22x^{-3}$

- The Difference Rule is used to find the derivative of a difference (subtraction) of functions
- The Difference Rule says that the derivative of a subtraction of functions is equal to the subtraction of each derivative
- When finding the derivative of a function that has additions ($+$) and subtractions ($-$), you need to get each derivative and keep the original sign for each term. If the exponent that comes down multiplying is negative, the sign of the term changes

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