# 1. The Differential

Earlier in the differentiation chapter, we wrote `dy/dx` and `f^'(x)` to mean the same thing. We used `d/dx` as an **operator.**

We now see a different way to write, and to think about, the derivative.

### Definition

### Mini-Lecture

See the

mini-lecture on differentials.

The **differential** of
`y = f(x)` is written:

`dy = f^'(x)dx`

**Note:** We are now treating `dy/dx` as a **fraction**, rather than as an **operator**.

### Example

Find the differential of `y = 3x^5- x`.

### Flash demonstration

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**Note**

We could use the differential to estimate the
real change in value of a function (`Δy`) caused by a small
change in `x`(written as `Δx`)*.* Many text books
do this, but it is pretty silly, since we can easily find the** exact **change - why approximate it?

We are introducing differentials here as an introduction to
the notation used in **integration**.

**How are*** dy, dx*
**and **Δ*y*
**and ****Δ***x* **related?**

*dy, dx*

*y*

*x*

As `Delta x` gets smaller, the ratio `(Delta y)/(Delta x)` becomes closer to the "instantaneous" ratio `dy/dx`.

That is, `lim_(Delta x->0) (Delta y)/(Delta x)=dy/dx`

See Slope of a tangent for some background on this.

We now go on to see how the differential is used to perform the opposite process of differentiation, which first we'll call **antidifferentiation**, and later **integration**.

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