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Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Video Transcript

So now that we've finished our topic on limits, we've built up all of the machinery that we need to go back to. The question of finding slopes of functions are slopes of the graphs of functions or, in other words, finding instantaneous rates of change. And now another word for this that will use from here on out is the derivative. And put simply, the derivative really is just the instantaneous rate of change at the function or the slope of the tangent line at a point. So let's just draw a picture just to remind you of the set up. So we have a function that I'll sketch. Maybe it looks something like that. And so if I have a point, X equals a then what I really am looking for is this tangent line here. Now there's a question about whether or not this tangent line exists that will sort of discuss. But if it exists, this tangent line will come and touch the graph of the function, and exactly one point so specifically it will touch the function at this point. A. That's a bit so. The derivative is nothing more than the slope of this line. So this line is gonna have an equation y equals mx plus B. And by the derivative, we really just mean the slope. Yeah, And now that we again, we have this machinery of limits to be able to take the limit of secret lines and tangents really are just the limit of secret lines. And that's what we're going to be doing in this topic. So the first thing we're going to dio is we're just going to define the derivative. We're going to very precisely say what the derivative is in terms of a limit. Now that we understand limits, weaken, define the derivative, then we're just going to practice finding derivatives. And specifically, what I mean is we're going to be finding what's called derivative functions. So we're gonna have a function, and then we're going to be able to find another function that the outputs of that function are actually the derivatives, the slope of the function itself or specifically, the slope of the tangent minds. And we're going to do this with all different types of functions. Polynomial is rational functions, Trina Metric functions, exponential functions, etcetera. We're just going to cover everything and then once we do that, we're going to talk about some applications and specifically what I mean by applications here is applications of the derivative or the instantaneous rate of change of a function at a point. And now these will span. All different areas of science will probably bend a little bit more towards the physical applications because they're very concrete, but will also touch on a few other types of problems that occur in economics and biology. Stuff like that. So you're going to see we're going to define the derivative, and you're going to see that it's an extremely useful and applicable concept, really across the board, in science and just in the way that our world works. And again, it's going back to this key idea about calculus. The calculus really is about the study of change of a function, and the derivative is giving us a that first tool of saying, okay, this is how a function is changing point by point