Subject: Differentiating...
Author:
Posted on: 2012-06-29 00:30:00 UTC
All right then, mini calculus lesson.
Differentiating is essentially finding the slope of a line or curve. Suppose you've plotted Y = 9X on a graph. You will get this nice straight line with a slope of exactly 9. That slope is an indicator of how fast your Y-value is changing: in this case, it is increasing at a rate of 9.
Now plot Y = X^2 and take X=2. Your coordinate pair is (2,4). You want to find the slope of X^2 at X=2. The first time I saw this, I thought: "But it's a curve! How am I supposed to find the slope of a curve?"
Now, the interesting stuff starts: try finding the slope of the line between X=3 and X=1. The answer is a slope of 4. Repeat it again using smaller increments closer to X=2: X=1.5 and X=2.5, X=1.75 and X= 2.25, etc... Now, you'll realize that you can't find it precisely at X=2 because the values would cancel each other out (a single point cannot have a slope), but you can get really, really close to the exact value, and that value is 4. If I take X=3 and do the same thing to it, I'll always end up with a slope of 6. If I take X=4, I'll get a slope of 8.
Do you see a pattern emerging here? The slopes of X^2 are determined by the function 2X. I have just differentiated X^2 and can predict the slope of any point on the X^2 curve! This means that I know how fast the Y-value is changing at a precise X-value, and it applies anywhere within the function's domain.
If I take X^3, I'll find that the slope of the X^3 curve is predicted by 3X^2. If I take X^4, I get 4X^3. See another pattern?
I have just demonstrated the power rule to you: the derivative of any function raised to a power can be represented as such:
Derivative of X^n is nX^(n-1).
This is but one of the many rules of differentiation but just as long as you remember that you are looking for the slope of an equation, it'll make a lot more sense.
A practical example: a car's position can be plotted using D= t^2. Find velocity and acceleration at any time t if speed at t=0 is 0 m/s.
1st derivative of position is speed: d/dt= 2t m/s.
2d derivative of position is acceleration (differentiate speed): d^2/dt^2= 2 m/s^2.