Calculator inputs
Enter derivatives at x = 0. The page builds a truncated Maclaurin polynomial, differentiates it to the chosen order, and evaluates it.
Example data table
This sample uses f(x) = ex, where every derivative at the origin equals 1. With N = 6 and m = 1, the derivative estimate at x = 0.5 approaches e0.5.
| n | f(n)(0) | Coefficient after differentiating once | Power used |
|---|---|---|---|
| 1 | 1 | 1 / 0! = 1 | x0 |
| 2 | 1 | 1 / 1! = 1 | x1 |
| 3 | 1 | 1 / 2! = 0.5 | x2 |
| 4 | 1 | 1 / 3! = 0.166667 | x3 |
| 5 | 1 | 1 / 4! = 0.041667 | x4 |
| 6 | 1 | 1 / 5! = 0.008333 | x5 |
Formula used
Original truncated Maclaurin series
PN(x) = ∑n=0N [f(n)(0) / n!] xn
m-th derivative of the truncated series
PN(m)(x) = ∑n=mN [f(n)(0) / (n - m)!] xn-m
Each table row shows one contribution to the chosen derivative order. The calculator sums all contributions to produce the approximation.
How to use this calculator
- Enter the function label so the result reads clearly.
- Type the evaluation point x where you want the derivative estimate.
- Select the highest Maclaurin order N available from your known derivatives.
- Select the derivative order m you want to estimate.
- Fill in f(0), f′(0), f″(0), and higher derivatives at zero.
- Optionally add the exact derivative value for error measurement.
- Press calculate to show the result above the form.
- Download CSV or PDF for reporting, class notes, or review.
Frequently asked questions
1. What does this calculator estimate?
It estimates a selected derivative of a function by differentiating a truncated Maclaurin polynomial built from derivatives known at the origin.
2. What inputs do I need?
You need derivatives at x = 0, an evaluation point, the truncation order, and the derivative order. An exact value is optional.
3. Why must the derivative order stay within the series order?
A truncated polynomial of order N cannot reliably produce a derivative order higher than N. Beyond that, every remaining term disappears.
4. How is the next omitted term estimate useful?
It gives a quick sense of how much the next missing series term might change the result. Smaller omitted terms usually suggest better truncation accuracy.
5. Can I use this for non-elementary functions?
Yes. If you know derivatives at the origin or obtained them from another method, the calculator can still build and differentiate the truncated series.
6. Does this replace an exact derivative?
No. It gives a series-based approximation. Accuracy depends on the truncation order, the evaluation point, and how strongly later terms influence the function.
7. Why do results worsen far from the origin?
Maclaurin polynomials are centered at zero. As x moves farther away, neglected higher-order terms can matter more, reducing approximation quality.
8. When should I increase the truncation order?
Increase it when you need better accuracy, especially if x is not very close to zero or the omitted term estimate is still large.