In calculus, **differential approximation** (also called *approximation by differentials*) is a way to approximate the value of a function close to a known value. It is just another name for tangent line approximation. In other words, you could say “*use the tangent line to approximate a function*” or you could say “*use differentials to approximate a function*“; They mean the same thing.

## Differential Approximation Notation

Although tangent line approximation and differential approximation do the same thing, differential approximation uses different notation.

- Δx = change in x,
- Δy = change in y,
- dx = change in x (for the tangent line),
- dy = change in y (for the tangent line).

The image below shows where these pieces fit on a graph.

At first, you may find it a little challenging to see how all of the pieces fit together. But you don’t need to fully understand what they all mean to solve a differential approximation question: the solutions are quite formulaic, and all you really need to know is a couple of different substitutions.

For example, notice on the graph that Δx and dx are of equal lengths. That means you can substitute one expression for the other. The following expressions are also interchangeable:

f(x + Δx) = f(x) + Δy = f(x) + f′(x)Δx.

The next example should make it clear how these substitutions work.

## Differential Approximation Example

As a simple example, let’s say the function is f(x) = √(x) and the known value you want to approximate is √(10).

Step 1: Choose a value close to your known value. √(9) is very close to √(10).

Step 2: State Δx (change in the function value for x). Here, we’ve gone from 10 to 9 for our x-value, so Δx = 1.

Step 3: Find the first derivative for the function. For this example, differentiate the function with the power rule (because you can rewrite √(x) as x^{½}). This gives:

Step 4: Write out the known value as a sum of the function value at x, plus the change in the function value at x:

√(10) = f(x + Δx) ≈ f(x) + f′(x) Δ x (where ′ is prime notation for the derivative. This is where we are using differential approximation instead of the actual value).

Step 5: Substitute in your actual function and derivative:

√(10) = √(x) + Δx

Step 6: Plug in your values, and solve:

√(10) = √(9) + (1)

Giving √(10)= 3.1667, which is close to the actual answer (found with a calculator) of ≈ 3.1623.

## References

Department of Mathematics, University of Houston. 1. Differential Approximation (Tangent Line Approximation). Retrieved January 11, 2020 from:

https://online.math.uh.edu/apcalculus/Week03-PracticeProblems-VideoVersion.pdf

Stewart, S. (2009). Study Guide for Stewart’s Single Variable Calculus: Concepts and Contexts, 4th Edition. Cengage Learning.

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