7a. The Differential
Integration Mini Video Lecture
The differential is an essential piece of the integral calculus puzzle. This video explains what they are.
Related lessons on IntMath
This video explains this page:
We use the idea of differentials later, in:
Solving Differential Equations (DEs) (don't worry, you aren't expected to understand DEs until after you've studied integration)
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.