Approximating Function Values with Differentials

What Differentials Are For

Sometimes a function value is difficult to calculate exactly, but you can find a nearby function value that is easy. Differentials let you use the derivative to approximate the change in a function.

The key idea is simple: if is close to a value where the function is easy to evaluate, then the function can often be approximated using the tangent line at .

For a small change in the input, , the corresponding change in the output is approximated by . This gives the differential relationship:

Because , we can use this relationship to estimate a function value without evaluating the original function exactly.

The Differential Approximation Formula

Suppose and we want to approximate using a nearby, easier value .

First, define the change in the input:

Then calculate the differential:

Finally, approximate the new function value by adding the estimated change to the known function value:

This is the main formula to remember:

A Step-by-Step Method

Choose a nearby easy value. Pick close to the value you actually want, where is easy to calculate.
Find the change in the input. Calculate .
Find the derivative. Determine .
Evaluate the derivative at the easy value. Find .
Calculate the differential. Use .
Approximate the function value. Add the estimated change to the known function value using .

Worked Examples

Example 1: Approximating a Square Root

Use differentials to approximate .

Step 1: Let . Choose , because is easy to evaluate.

Step 2: Find the change in the input: .

Step 3: Differentiate: .

Step 4: Evaluate the derivative at : .

Step 5: Calculate the differential: .

Step 6: Approximate the function value: .

Result: .

Example 2: Approximating a Cube Root

Use differentials to approximate .

Step 1: Let . Choose , since is easy to evaluate.

Step 2: The change in the input is .

Step 3: Differentiate: .

Step 4: Evaluate at : .

Step 5: Calculate the differential: .

Step 6: Approximate the desired value: .

Result: .

Example 3: Approximating a Trigonometric Value

Use differentials to approximate .

Step 1: Let . Choose , because is easy to evaluate.

Step 2: Calculate the input change: .

Step 3: Differentiate: .

Step 4: Evaluate at : .

Step 5: Calculate the differential: .

Step 6: Approximate the function value: .

Result: .

Example 4: Approximating a Power

Use differentials to approximate .

Step 1: Let . Choose .

Step 2: The change in the input is .

Step 3: Differentiate: .

Step 4: Evaluate the derivative at : .

Step 5: Calculate the differential: .

Step 6: Approximate: .

Result: .

Why the Approximation Works

Differential approximation works because a differentiable function looks increasingly like its tangent line when you zoom in near a point.

The tangent-line approximation at is:

Since , this can also be written as:

The differential method is therefore really a tangent-line approximation. Differentials give us a convenient way to calculate the estimated change in the function.

Common Mistakes

Choosing a reference value that is not easy to evaluate

The entire point of differential approximation is to replace a difficult calculation with an easy one. For example, when estimating a square root near 25, use , not an arbitrary decimal such as .

Using the wrong change in x

The differential uses , where is the value you want and is your convenient reference value. Keep the direction of the change correct.

Forgetting to evaluate the derivative at the reference point

You need , not the derivative evaluated at the target value. The approximation is based on the tangent line at the nearby, easy point.

Confusing dy with the exact change in y

The differential is an approximation to the actual change . They are close when the change in is small, but they are not generally identical.

Practice Problems

Question 1

Use differentials to approximate .

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Question 2

Use differentials to approximate .

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Question 3

Use differentials to approximate .

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Question 4 (Conceptual)

Explain why is a reasonable approximation for when is close to .

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