# 7a. The Differential

## Integration Mini Video Lecture

The differential is an essential piece of the integral calculus puzzle. This video explains what they are.

## Video transcript

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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 3*x* to the power of 4, the derivative, *dy* over *dx* is equal to [four 3s are] 12*x* 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 12*x* 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 12*x* 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.

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