Derivatives …and understanding the meaning of a derivative at a point

(it’s really not that bad, So… what are we going to learn? 1. The meaning of a derivative 2. How derivatives are related to slope 3. Types of derivatives (it’s really not that bad, I promise)

So… what does this have to do with Calculus? Well, derivatives are really essential to Calculus. When you learn about integrals and other fun things you will need to base all of it on your knowledge of derivatives. (Also you need to be really good at them to pass Mr. Rahn’s Derivative test)

Definition of a Derivative at a point. What is a derivative? Well its actually really simple. A derivative is the term for the slope of a line or a curve at a point.

Let’s begin with a numerical example of how to find the derivative at a point. To the right is a graph of Now lets say that you wanted to find the slope of this equation at x=2. You would need to take the derivative which is and plug into this equation 2 for x and solve. The slope of y is? 4

Wait… We’re not finished yet!!! It’s time for an analytical description of a derivative at a point. (This is even easier, I promise) Remember y=mx+b? Well, the m in it is simply the term for slope. Its really the same thing as the derivative. Okay, so if you wanted to find the slope of anything at a point, the best way to do this is to take the derivative!!!!

Okay, so you get the basic meaning of a derivative at a point, but how can this apply to real life situations? Here’s an example, a runner is running on a very hilly road, as she goes uphill her velocity is low, however when she is running downhill her velocity increases dramatically. The sin curve to the right is a perfect example of the runners changing velocity.

Now if you were to take the derivative of a velocity graph, the derivative would show a graph of acceleration. The derivative of sin x is y= cos x. The graph of cosine of x is actually in this example the acceleration of the runner. Simple huh?

Now don’t let it stop here Now don’t let it stop here. The derivative of acceleration is called a jerk, and we could keep on going on and on. But the point is that derivatives can tell us a lot of important information about anything.

Now, let’s review. The most basic meaning of a derivative is that it is the slope of a line or a curve at a point. However, derivatives can also have different meanings. For example the derivative of the velocity of a particle actually produces a graph of the acceleration experienced by the particle.

All about me… I survived Mr. Rahn’s class, need I say more? Okay okay, I’m a junior at Southern Regional High School, and I have another year of Calculus with Mr. Rahn (yeah) and Spencer (gag).