So in recent videos, we got introduced to the idea of a derivative. And recall that a derivative is just the slope of a tangent line at a certain value, like x equals 1 or x equals negative 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 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.
So here we have this example where we're asked to find the derivative of this function, f(x)=x2, for any x value. And we're asked to use this to find the slope of the tangent line for these two specific values, x equals 1 and x equals negative 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. Now to do this, what I need to do is use the equation for a derivative. And something that I'll mention here is that 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. And the derivative of our function is going to be the limit as h approaches 0 for f(x+h)-f(x)h. So 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 right here is what exactly is this h that we're seeing? Right? Because this is not a variable we're really familiar with. Well, it turns out all this h is, is it's basically just taking the place of this x minus c that we saw in the previous equation, where we had two points and they slowly converged to one specific value that we were looking at. Well, 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. So we have our function f(x)=x2. And if I want to use this equation up here, well, what I'm going to do is plug in these variables. So what I see is that our function is x squared.
So f(x+h) is going to be x+h2. And then we're going to have minus f(x), which I can see is x squared, all divided by h. So 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 right 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 I need to find some sort of way to get this h to go away in the denominator. And to do this, well, I'm first going to start by factoring this expression right here. So we have x plus h squared, and that can also be written as x squared plus 2xh plus h squared, and then this is going to be minus this x squared we have here, all divided by h. So this portion right here is just me expanding x plus h squared out. But notice when we do this, I can actually get rid of this x squared that I see.
And I can also take this h and factor it out of each of these terms. So what I'm going to end up having is h times two x plus, and then we're going to have 1 h left over in the parenthesis here, all divided by h. Now this is a great position to be in because notice I can now cancel one of the h's. So all we're going to have is 2x+h. Now at this point, I can apply our limit by taking 0 and plugging it in for h.
That's going to give us 2x+0, which is simply equal to 2x. So this right here is the general equation for the derivative, and that would be the derivative at any possible x value right there. That's the solution. Now we're not done with this problem yet, because we're also asked to find what the slope of the tangent line is at x equals 1 and x equals negative 2. Now to do this, well, we just need to use this result that we got right here.
Because using this equation right here called the definition of the derivative, we were able to find the derivative which is the slope of the tangent line. So what I'm just going to do is take each of these values, x equals 1 and x equals negative 2, and plug it into this equation. So if I find f′(1), that's going to be evaluating where x is equal to negative one, so I just need to replace this x here with 1. So we're going to have 2 times 1, which is equal to 2. And that right there is the slope of the tangent line at x equals 1.
Now notice something. When we solved this type of problem before, the slope of our tangent line at x equals 1 was also 2 for the function x squared. But rather than having to go through this whole process of finding these two points, well, once we found the general derivative, we could just plug this number directly in, and it gave us the same result. But now let's try a different value, like x equals negative 2. Well, to find this, I just need to take this x right here and replace it with negative 2.
So we're going to have 2 times negative 2, which is equal to negative 4. So that right there is the slope of the tangent line at x equals negative 2. And so if we were to go onto our graph here and we were to draw a tangent line right about here, this tangent line would have a slope of negative 4. And that is what this equation does. This general equation allows us to find the derivative at any possible x value.
And the nice thing is we don't even have to stop here. If we wanted to, we could figure out what the slope of the tangent line is at x equals 100, or x equals 1000, or even x equals negative 56, or just something random. We could plug whatever x value we want into this equation right here, and it would give us the slope of the tangent line at that specific value. So that is how you can find the general equation for the derivative, as well as the derivative at certain x values using this definition right here. So I hope you found this video helpful, and let's try getting some more practice.
