7a. The Differential
Integration Mini Video Lecture
The differential is an essential piece of the integral calculus puzzle. This video explains what they are.
What is a differential?
Well its no big deal really. But it is important in the understanding of integration.
A differential is just like a derivative except we write it across the page.
Let me illustrate, if I start with y equals 3x to the power of 4, the derivative, dy over dx is equal to [four 3s are] 12x to the power 3. `(dy/dx = 12x^3)`
Now this is a derivative, to write it as a differential, all I have to do is this: dy equals 12 x cubed dx. And I've written it in this form, dy equals the derivative expression 12x cubed dx. `(dy = 12x^3dx)`
That's all there is to it really, what we do later is we use it in integration.
What you'll see later is we'll have the integral of say 12x cubed dx. `(int 12x^3dx)`
This is the proper way to write an integral, integral sign, and then a differential. Hopefully it will become clearer the more examples that you see later.