# 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.

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.

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, dxand Δyand Δxrelated?

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.

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