Finding Derivatives of Sums, Products, Differences & Quotients

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will go over how to find the derivative of a sum, difference, product, and quotient. We will look at the different formulas involved in these derivatives and use those formulas to calculate some derivatives.

Working with Derivatives

Suppose you have a job for a company that builds cars. Your boss tells you they are working on a car part, and a certain aspect of it can be represented by one of the following four functions.


sumdiff1


You're told that you need to be able to find the slope of each of these functions for any given value of x. You remember that the derivative of a function represents the slope of that function at any given point. Great! All you have to do is find the derivative of each of these functions and your boss will be a happy camper.

You look at each of the functions and realize that the first function is a sum of functions, the second function is a difference of functions, the third function is a product of functions, and the fourth function is a quotient of functions. Thus, we want to look at how to find the derivative of a sum, difference, product and quotient. Let's get started!

Derivative of a Sum

When calculating the derivative of a sum, we simply take the sum of the derivatives. This is illustrated in the following formula.


sumdiff2


The first function that your boss wants you to work with is the sum of two functions. Therefore, to find the derivative of this function, we just take the sum of the derivatives. To do this, we need to recognize that the derivative of x 2 is 2x, and the derivative of 4x is 4. Now we just plug into our formula and we've got our derivative.


sumdiff3


We see that the derivative of the first function is f ' (x) = 2x + 4.

Derivative of a Difference

The second function, g, is a difference of functions. When we want to find the derivative of a difference, we simply find the difference of the derivatives. This is similar to the sum rule, and is shown in the following formula.


sumdiff4


To use our formula for our example, we just need to know that the derivative of 3x3 is 9x 2, and the derivative of 5x 2 is 10x. Once again, we just plug into our formula.


sumdiff5


Using our formula, we've found that the derivative of the second function is g' (x) = 9x 2 - 10x.

Derivative of a Product

Now things get a little trickier. The third function is a product of the functions x and ln(x). When we want to find the derivative of a product, we use the product rule for derivatives.


sumdiff6


Thus, to find the derivative of our function h in our example, we need to know that the derivative of x is 1, and the derivative of ln(x) is 1/x, and then we just need to use our formula.


sumdiff7


We see that the derivative of the third function is h' (x) = ln(x) + 1.

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