1. The Differential

Earlier in the differentiation chapter, we wrote math expression and f '(x) to mean the same thing. We used math expression as an operator.

We now see a different way to write, and to think about, the derivative.


Definition: The differential of y = f(x) is written: dy = f '(x)dx.

Mini-Lecture

See the
mini-lecture on differentials.

Note: We are now treating math expression as a fraction.



Example:

Find the differential of y = 3x5- x.

 

Answer

dy = f'(x)dx

dy = (15x4 - 1)dx

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?

math expression

See Slope of a tangent for some background on this.


Relative Error

Note: We may perform a calculation and find an error of 1 cm. If this is part of a problem involving lengths of 8 cm, then we have a big problem. If it involves lengths of 3 km, we have an insignificant problem. We usually talk about the relative error:

math expression or math expression

rather than the absolute error of 1 cm.

 

Proportionate and Percentage Change

The proportionate change in y(x) is given by math expression (no units).

The percentage change in y(x) is given by math expression



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