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 Δy and Δx related?

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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