Linearity of the Derivative

In the previous sections, we discussed the concept of the derivative of a function and how it can be used to describe the rate of change of the function at a specific point. In this section, we will explore some properties of the derivative that are crucial in calculus. Specifically, we will discuss the linearity of the derivative.

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Adding Functions

Adding Functions

Suppose we have two functions, and , and we want to find the derivative of their sum, .

Recall that the derivative is essentially a limit of an average rate of change, which is a ratio of the change of the function to the change in the input. Therefore, we can write the derivative of as:

So, the question is, what is ? Well, it is simply the sum of the changes in and : .

One can easily show this by expanding the expression:

\begin{aligned} ended with \end{equation}

However, we can also see this on a graph. Let's consider the graph of and , as well as their sum:

Table of Contents​

Adding Functions​

Table of Contents

Adding Functions