Posted: Feb 16, 2012 11:28 am
by Darkchilde
I am starting a thread on Differentiation. Please do not post here, but use the discussion thread for any questions or comments you may have. This will be about differentiating any equation; we will start with understanding what differentiation represents in mathematics and in physics. This will be non-technical, so for the mathematically inclined, I will not use limits for the moment.

Let's get a simple equation of a parabola:

[math]


This translates to the following image:



a8692f6d28486eddebd317d1bc0939a4-2.png


a8692f6d28486eddebd317d1bc0939a4-2.png (18.57 KiB) Viewed 8705 times





The derivative of this is :

[math]


Mathematically, the derivative gives the slope, that is the inclination, of the line that is tangent at point x. I will show you an example of this with the above, so you can see what I am talking about. The ' after f, that is f' means the first derivative. There are a couple of notations used in physics and mathematics on this one, but I will explain them as I use them. So the second derivative will be noted as f''(x), the third one as f'''(x); however after the 3rd derivative we mostly use superscript numbers like this: f4(x)

Now, one point of the parabola is (1,1), that is the coordinates are x=1, and y=1. Coordinates always are written in the form (x,y). I take this point because it is very easy to calculate.

Now, a line has equation

[math]


where m=slope, and c=intercept with the vertical (y) axis in the graph. Remember that the horizontal axis, is called the x-axis, and the vertical axis is called the y-axis.

So, we need to find the tangent to the parabola at the point (1,1). Then we can find m, the slope from:

[math]


Putting x=1 as is the x-coordinate of our point:

[math]


And so the slope m=2. Now to find the intercept c, by substituting the coordinates of the point (1,1) in the equation of a line:

[math]

[math]

[math]


And so we have the equation of the line:

[math]


If we put this into the first graph we get:



plot2.png


plot2.png (22.09 KiB) Viewed 8705 times





As you can see the line touches the parabola, a curve in just one point: (1,1).

In physics, a derivative shows the rate of change of a quantity; for example, if you look in the equations thread in Physics, you will see that the derivative of position, is velocity. Velocity is the rate of change of position; and similarly, the derivative of velocity is acceleration, which is the rate of change of velocity.

In physics, and other sciences, most times we need to see how the rate of change of a quantity behaves; velocity may not behave linearly for example, and the derivative function shows how velocity changes, and what we can expect.

Tomorrow I will start with the simple rules of differentiation: constants and powers.