The Geometric Meaning of Differentials

Over my last 3 posts, we've learned to construct a tangent line L(x) on any given point on a differentiable curve. The picture below shows the geometric meaning of a differential.

Figure 1.
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Now to explain what it means. Suppose we have the function...

We note that y is the dependent variable and x is the independent variable which can be any real number.

If f(x) is a differentiable function, then the differential dy can be written in terms of the differential dx as...

Thus dy is also a dependent variable, since it depends on the values of x and dx. What the graph shows above is that dy measures the rise or fall of the tangent line L(x) at a given point along the curve.

However, what we can see is in the picture above, given the distance of dx, there is quite a large discrepancy between the rise of the tangent line dy and the rise of the curve f(x).

A more detailed picture of this is shown below, where Δy is the rise of the curve.

Figure 2.
Created with: www.desmos.com/calculator

Now, from Figure 2, the coordinates of the point Q are:

The coordinates of the point P are simply:

Thus the vertical distance between Q and P then is the rise of the curve Δy...

As you can see from the picture, the difference between Δy and dy become smaller and smaller as Δx or dx decreases.

In the next tutorial, we will compare Δy and dy for a defined function.

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Here's a list of posts created so far on the subject of Linear Approximations and Differentials:

1. Linear Approximations Part 1 - Interpolating between Empirical Data
2. Linear Approximations Part 2 - Estimating values of f(x) = √(x+2) near x = 2
3. Linear Approximations Part 3 - why sin(x) ≅ x near 0
4. The Geometric Meaning of Differentials

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