# 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.

### 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.

### 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, dxand Δyand Δxrelated?

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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