# What is A Derivative?

Discussion in 'High School' started by mathman, Sep 25, 2007.

1. ### mathmanRookie

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Sep 25, 2007

First, you need to think of a derivative as measuring the rate at which something changes as measured by something else. We do that a lot in life and never think of it as a derivative. For example, in a car, the speedometer is a measure of the change in position per unit of time. In this case we would write dP/dt where P stands for position along some path of travel (in miles) and t stands for time (in hours).

Of course the speedometer only works correctly when you are going
forward. In airplanes they have a gauge that measures the same thing for elevation. It measures how fast you are going up (or down), which is important as you can imagine. When you see the stock market report on the TV in the evening, it measures the change in the aggregate stock index per unit of time. In
this case the units are dollars per day.

On a graph, the visual pattern is one of slope. If we measure the
closing stock price from day to day we notice that the graph gets
higher on days when the price change is positive, and lower when the stocks go down. The steeper the slope, the faster the change, and positive means I'm making money.

This makes sense in terms of how the derivative is defined. The basic part of the formula for the derivative is just the formula for slope. The instantaneous part is where the limit notation comes in. Let's look at something simple like y = x^2.

If we wanted to find the derivative at x = 3, we could look first at
the graph for a clue. Is the curve going up or down? Imagine a tangent to the curve at x = 3. The slope of the tangent line is the slope of the curve at that point, by definition.

We find it numerically by taking two points, x1 and x2, and the
associated y values y1 and y2. In our case x1 = 3 and y1 = 9. If we
pick x2 to the right of x1, then the slope between the two points
would be given by:

m = (y2 - 9)/(x2 - 3)

To find the "instantaneous" change, we just let x2 get closer and
closer to 3. This is where the limit part comes in: as x2 goes to 3,
the change in x (which we call dx) goes toward zero.

Last edited: Sep 25, 2007
2. 3. ### AliceaccMultitudinous

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Sep 25, 2007

Again, I'm not sure of your point.

I define derivative as "the slope of the tangent line to a function at a particular point." I realize that the first derivative can also be defined as velocity, and the second as acceleration.

But why post this in a teacher's forum? I'm not knocking you for it, I'm just not sure why this thread is here.

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Sep 25, 2007

WHAT?? Are you asking a question??

5. ### TeacherGroupieModerator

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Sep 25, 2007

This has me puzzled too.

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