Differentials

#folder

Assuming is small, we know that $$\frac{\Delta y}{\Delta x} \approx \frac{d}{dx}$$
This means we can write , and as a result

Note that this is just "abstract nonsense", since only make sense as a quotient.

Definition

The differential of is written

Tangent Line Approximation

Tangent Line Approximation

The origin of many approximation methods is Newton's Quotient:

Newton realized that you can use this practically. If is small,

Or,

Setting :

Equation

A linear approximation is defined by:

Intuition

We can intuitively make sense of this if we want. We have the point of the function we know, and then add the rate of change of the function at that point, and follow that rate until we reach the x-coordinate we want.

Visualization

Example

Approximate

solution
Knowing , let . We want .
We will try a linear approximation. Let , which is close to 30. and are simple to calculate.


So,

Compare this to the actual value of

Equation

If we take the approximation, we can find the differential to use:

So

Example

Use differentials to approximate

solution
Knowing , we identify , and .
We then calculate

When decreases by , decreases by approximately . Thus

Compare this to the actual value of

Remark

Interesting note: L'Hôpital's rule comes from the linear approximation