# 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.

We will use this new form of the derivative throughout this chapter on Integration.

### Mini-Lecture

See the

mini-lecture on differentials.

### Definitions

**Differentials** are infinitely small quantities. We usually write differentials as `dx,` `dy,` `dt` (and so on), where:

`dx` is an infinitely small change in `x`;

`dy` is an infinitely small change in `y`; and

`dt` is an infinitely small change in `t`.

When comparing small changes in quantities that are related to each other (like in the case where `y` is some function f `x`, we say the **differential** `dy`, of
`y = f(x)` is written:

`dy = f'(x)dx`

**Note:** We are now treating `dy/dx` more like a **fraction** (where we can manipulate the parts separately), rather than as an **operator**.

### Example 1

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

Answer

Here `y=3x^5-x`, so `f'(x)=15x^4-1`.

So the differential is given by:

`dy = f'(x)dx`

` = (15x^4-1)dx`

To find the differential `dy`, we just need to find the derivative and write it with `dx` on the right.

### Example 2

Find the differential `dy` of the function `y = 5x^2-4x+2`.

Answer

Since `y=5x^2−4x+2`, then `f'(x)=10x-4`.

So the differential is given by:

`dy = (10x-4)dx`

**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*

`Delta y` means "change in `y`, and `Delta x` means "change in `x`".

We learned before in the Differentiation chapter that the slope of a curve at point *P* is given by `dy/dx.`

The slope of the dashed line is given by the ratio `(Delta y)/(Delta x).` As `Delta x` gets smaller, that slope becomes closer to the actual slope at *P*, which is 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**.