So in recent videos, we were introduced to the idea of a derivative. Recall that a derivative is just the slope of a tangent line at a certain value, like x=1 or x=−2. Now in this video, we're going to learn how we can find this derivative, also known as the slope of a tangent line, at any possible x value at any possible point. And that might sound a bit scary, but it turns out this process for finding the general equation of a derivative is actually pretty similar to the process we took for finding the slope of a tangent line. So without further ado, let's just jump right into an example of what one of these problems might look like in this course.
Here we have this example where we're asked to find the derivative of this function, f(x)=x2, for any x value. We're asked to use this to find the slope of the tangent line for these two specific values, x=1 and x=−2. Now our first step when solving these types of problems is to find the general equation because that's going to give us the slope of the tangent line for any possible x. To do this, what I need to do is use the equation for a derivative. Whenever you see derivatives, they could be written in this notation.
These are all types of things you could see in your textbook. This little apostrophe here is what we call prime. So if I want to find the derivative of our function, it's going to be f′(x). The derivative of our function is going to be the limit as h approaches 0 of fx+h−f(x)h. This is the equation you're going to need, and you may want to write this down.
Now a question that will come up about this formula is what exactly is this h that we're seeing? Because this is not a variable we're really familiar with. Well, it turns out this h is basically just taking the place of this x−c that we saw in the previous equation, where we had two points that slowly converged to one specific value that we were looking at. In this equation, since we're trying to find the slope of a tangent line for any possible x, this h here is any possible two points.
And since we know they're going to get really close to converge at one value, we see this limit of h approaches 0. So let's go ahead and see if we can use this equation to find the general equation for a derivative. We have our function f(x)=x2. To use this equation, what I'm going to do is plug in these variables. Our function is x2.
So f(x+h) is going to be (x+h)2, and then we're going to have minus fx, which is x2, all divided by h. That right there is using this equation and applying it to this example.
Now what I can try to do is evaluate this limit, but I noticed that we have h approaching 0. If I try to put 0 in there, we're going to be dividing by 0, which is a math rule that we cannot break. So what I need to do here is find some sort of way to get this
