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I am a 9th grade student in a college prep program, and I have started to learn Derivatives, and Integrals. (I am taking Algebra II Honors PIB and Geometry Honors PIB right now) I can derive very well (for a 9th grader with no formal instruction), but I only have a faint idea of how these derivatives are useful. How do are they used? Examples are appreciated
I'll give you an example:
Let's say that a business must buy certain materials in order to keep operations going. Let's also say that the cost of buying these materials varies during different months of the year according to the function
C=x^2-2x +1. At which month would the cost of materials be at a minimum?
Hint: You're looking at x on the interval (0,12].
If you need help with this problem let me know.
Use the derivative to find the answer! Not the graph!
So why even bother? =pEverything that can be said on this subject will have been said many times elsewhere - use Google.
Anyway, a very common example is derivation of a position-time graph. Let's say we have some distance R that can be found using the equation 2t^3 - 3t^2 + 2. By finding the first and second derivatives, we can learn a lot more about the object travelling this path than that it moved 2 units (we'll just say meters) at 0 seconds.
R' = 6t^2 - 6t
This is dR/dt, our change in distance divided by change in time. You may recognize it as velocity (without getting technical). So, now we not only know how far the object has travelled after x seconds, but we also know how fast it was going at x seconds.
Now if we derive once more, we'll get d(dR/dt)/dt. Which you may recognize as acceleration.
R'' = 12t - 6
What does this tell us? It tells us that our acceleration is constantly changing. But we can also pull something else from this; maximum and minimum velocity. Tell me, how would you identify a maximum velocity? It'd be one of the times the object stops accelerating, right? Surely, if the object stops accelerating at a point, than after that point it must either go up, down, or stay the same.
So set R'' = 0. When does this occur? When t is 1/2, right? So the object's acceleration is 0 at 1/2 seconds. But, that doesn't tell us whether we're dealing with a min or max. How do we tell? Well, what's happening to the left of the point? What's happening to the right?
At 0, our R'' is -6. So our acceleration is going down. At 1, our R'' is 6, meaning it's going back up again. This tells us that our absolute minimum value for R' (dR/dt) occurs at 1/2 seconds.
For example, the derivative of a linear function y = mx + b, gives you the slope, which is the rate of change of the line
For a quadratic function, a parabola for example, its derivative gives you the equation of a tangent line to the parabola at a particular point.
The derivative of a position function (i.e. where an object is at a certain time) is velocity, the instantaneous rate of change of the object. Then, the derivative the velocity is acceleration, the change in velocity over time.
So, generally, a derivative can calculate rates of change.
Hope that helps! Derivatives are FUN!!!!! It will make your mathematical life MUCH easier! Good luck!