Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
Introduction to Derivatives
Let's take a look at Super C. Super C's height as a function of time looks a little bit like this. We know that Super C's velocity slows down as soon as he is launched from the cannon - at least his velocity in the vertical direction. His velocity at any given point in time is his instantaneous rate of change. Well, in calculus, we call that the derivative. The derivative is a function for the instantaneous rate of change. So how do we find this derivative? Well, what is the instantaneous rate of change? We know that the average velocity, the average rate of change, is the end point minus the start point, divided by the region: (f(b) - f(a))/(b - a). So this is the change in location divided by the change in time.
Well, that's the average over some long time period. But maybe we should look at it over a very short time period. Maybe, to make this an instantaneous rate of change, that time period is really zero. To make this an instantaneous rate of change, we're going to look at the limit as the change in time goes to zero - so, as our little region of time that we care about goes to zero.
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Let's write this a little more formally. In terms of position as a function of time, x sub end is really some function of t where t is going to be equal to t + delta t. x sub start is the function at t. So all this is saying is that we're evaluating our function both at the begin time, t, and at the end time, t plus whatever our interval size is. So putting this all together, I'm going to look at the limit as delta t goes to zero - so as our interval goes to zero - of our end point, which is x sub end, f(t + delta t), minus our start point, x sub start, which is f(t), all divided by delta t. And this is one of the most important parts of calculus, the derivative. Specifically, this is the derivative of f(t).
Now we can generalize this to a function y=f(x). For y=f(x), formally, the derivative is written as the limit as delta x goes to zero - so this is our region going to zero - of one end (f(x + delta x)) minus the other end (f(x)), all divided by delta x. So this is the derivative. We will also call this y`, or f`(x), or dy/dx, or df/dx, or d/dx(f(x)). There are a lot of names for this, but really, all it means is the derivative of y=f(x). My personal favorite is y`, but sometimes we have to be very specific and say dy/dx, which is the derivative of y with respect to x.
So let's recap. The derivative is the rate of change at a single instance. We write that as the limit of some interval going to zero of f(x) at one end of the interval minus f(x) at the other end of the interval, all divided by the interval length.
After watching this lesson, you should be able to define derivative and write the equation to find it at begin and end times, as well as its relation to instantaneous rate of change.
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Derivatives: The Formal Definition
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