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.

Definitions

Mini-Lecture

See the
mini-lecture on differentials.

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`.

Example 2

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

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 Δy and Δx related?

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

Relationship dx, dy, delta x, delta y

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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