Sum Rule for Business Calculus

with sample problems

Intro

When you have two or more functions, you can perform operations between them, such as adding them, subtracting them, dividing them, etc. We will start with the concept of differentiating functions that are adding.

The Sum Rule is used to find the derivative of a function that contains a sum of other functions.


When to Use

Here are some examples of functions that are equal to the sum of other functions.

$$f(x)=4x^4+3x^2$$

$$f(x)=x+1+5x^{64}$$

$$f(x)=\sqrt{x^3}+2x^4+10,000$$

What all these functions $f(x)$ have in common is that they are equal to a sum of other functions. For all these cases, we apply the Sum Rule.


How to Apply

Below, you will learn how to apply the Sum Rule.

sum rule for business calculus

Learn the concept

In words: The derivative of a sum of functions is equal to the sum of the individual derivative of each function

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

Pretty straightforward, right? Let's look at an example below.


Sample problem

Given that $f(x)=2x^3+5x^2$,

Question 1Find $f'(x)$

The function $f(x)$ is the sum of two functions, which are $2x^3$ and $5x^2$. These two functions are adding, as evidenced by the plus sign between them. To find the derivative of this function, we use the Sum Rule. To do so, we get the derivative of both functions and add those derivatives together:

$$f'(x)=(2x^3)'+(5x^2)'$$

We apply the Power Rule to get each derivative,

$$f'(x)=3 \cdot 2x^{(3-1)}+2 \cdot 5x^{(2-1)}$$

After simplyifing the above, we get our final answer,

$$f'(x)=6x^2 + 10x$$

As you can see, we got the derivative of each term in the sum, and added the two derivatives together. If you have problems understanding how these derivatives were obtained, please visit the Power Rule section.

But functions can be composed of more than two functions adding. Let's look at another example below.


Sample problem

Given that $f(x)=5x^{10}+30x^2+2x+1,000$,

Question 1Find $f'(x)$

Here, since these $4$ functions are adding, we can use the Sum Rule. We get the derivative of each function individually, and add them up,

$$f'(x)=10 \cdot 5x^{(10-1)} + 2 \cdot 30x^{(2-1)}$$

$$+ 1 \cdot 2x^{(1-1)} + 0$$

(if you are wondering why the last term is $0$, it's because $1,000$ is a constant, and it's derivative is zero, as per the Constant Rule)

We multiply coefficients and simplify exponents,

$$f'(x)=50x^9+60x^1+2x^0$$

And we simply to get our final answer,

$$f'(x)=50x^9+60x+2$$


Lesson Review

  • A function can be composed of the sum of other functions
  • The Sum Rule is used to find the derivative of a sum of functions
  • The Sum Rule says that the derivative of a sum is equal to the sum of each individual derivative

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