# Using sum difference derivative instead of prodict rule?

• Nano-Passion
In summary, the conversation discusses the different approaches to multiplying and simplifying the expression f(x)=(x^3-x)(x^2+2)(x^2+x-1). One approach involves using the sum rule, while the other involves using the product rule twice. However, it is mentioned that since the conversation is about finding the derivative, there is no need to multiply the expression at all. Both approaches will result in the same answer, so the simpler approach should be chosen.

#### Nano-Passion

Let us say we have this:

$$f(x)=(x^3-x)(x^2+2)(x^2+x-1)$$

Would it be equally correct to multiply the first two binomials, and then taking that and multiplying by the last tri-nomial; and then using the sum rule? It seems perfectly fine (and much simpler!) but I want to make sure I am not violating anything here. It seems like a longer problem to use the product rule twice.

Either way will work. Just guess what is most simple and start in.

Since you are talking about the derivative, I see no reason to multiply it out at all:
(fgh)'= fgh'+ fg'h+ f'gh.
$$((x^3−x)(x^2+2)(x^2+x−1))'= (x^3- x)(x^2+2)(2x+1)+ (x^3-x)(2x)(x^2+X-1)+ (3x-1)(x^2+2)(x^2+x-1)$$

HallsofIvy said:
Since you are talking about the derivative, I see no reason to multiply it out at all:
(fgh)'= fgh'+ fg'h+ f'gh.
$$((x^3−x)(x^2+2)(x^2+x−1))'= (x^3- x)(x^2+2)(2x+1)+ (x^3-x)(2x)(x^2+X-1)+ (3x-1)(x^2+2)(x^2+x-1)$$
True, I just figured I should simplify that more, so I wanted to do the other route. But now I see you don't have to simplify that more.

lurflurf said:
Either way will work. Just guess what is most simple and start in.

Okay, never hurts to be completely sure. :)

## 1. What is the sum difference derivative and how does it differ from the product rule?

The sum difference derivative is a mathematical method used to calculate the derivative of two functions that are added or subtracted from each other. It differs from the product rule, which is used to calculate the derivative of two functions that are multiplied together.

## 2. When should the sum difference derivative be used instead of the product rule?

The sum difference derivative should be used when the functions being differentiated are added or subtracted from each other, rather than multiplied. This method is particularly useful for simplifying complex expressions.

## 3. How is the sum difference derivative calculated?

To calculate the sum difference derivative, you take the derivative of each function separately and then add or subtract the results, depending on whether the functions are being added or subtracted. For example, if you have two functions f(x) and g(x), the sum difference derivative would be f'(x) ± g'(x) depending on whether the functions are being added or subtracted.

## 4. Can the sum difference derivative be applied to more than two functions?

Yes, the sum difference derivative can be applied to any number of functions that are added or subtracted from each other. The derivative of each function is still calculated separately and then added or subtracted accordingly.

## 5. What are the advantages of using the sum difference derivative?

One advantage of using the sum difference derivative is that it can simplify complex expressions and make them easier to differentiate. It can also be helpful in finding derivatives of multi-variable functions. Additionally, the sum difference derivative can be used to find the derivatives of functions that cannot be differentiated using the product rule, such as trigonometric functions.