In this article, we will first quickly summarize what it means to find the derivative of a function with a few examples. After we will show how we can prove that a function is differentiable for every x equal to a real number, R. To do so, we will need to use the concept of continuity.
Recap of the Derivative
We remember that the derivative is the rate of change of a function with respect to its variable, x. In other words, the derivative simply shows how much the function is changing, either increasing or decreasing, in a given point. Let us see a quick illustrative example. Imagine that we have the following function:
We can also see the graph which looks like so:
Then, for example, if we take the derivative at x = 4, then it’s simply how much the graph is decreasing at that point. We can see it would be equal to:
Proof of Differentiability at a Point
This is the section where we will need to use the concept of continuity. This is since we have the following fact:
Let us try to see this fact illustrated visually, so we can see why it’s important to understand this. We see:
