Calculator Inputs
Enter Taylor coefficients for f(x) = Σ cn(x-a)n. The calculator differentiates the truncated series around center a.
Formula Used
Assume the function is represented by a Taylor series around the center a:
The k-th derivative of the truncated series is:
The derivative at the center becomes especially simple:
This is why Taylor coefficients are powerful: each derivative at the expansion center is directly tied to one coefficient.
How to Use This Calculator
- Select a preset function or keep the form on custom mode.
- Enter the expansion center a and the evaluation point x.
- Choose the derivative order k you want to compute.
- Enter coefficients c0 through c10 for the truncated Taylor series.
- Set a graph range to compare the original series and its derivative visually.
- Press Calculate Derivative to show the result above the form.
- Use the CSV or PDF buttons to export your calculated summary.
Example Data Table
Example coefficients for the Maclaurin series of e^x up to order 10.
| Power n | Coefficient cn | Series term | Derivative at center n!cn |
|---|---|---|---|
| 0 | 1 | 1 × x^0 | 1 |
| 1 | 1 | 1 × x^1 | 1 |
| 2 | 0.5 | 0.5 × x^2 | 1 |
| 3 | 0.1666666667 | 0.1666666667 × x^3 | 1 |
| 4 | 0.0416666667 | 0.0416666667 × x^4 | 1 |
| 5 | 0.0083333333 | 0.0083333333 × x^5 | 1 |
| 6 | 0.0013888889 | 0.0013888889 × x^6 | 1 |
| 7 | 0.0001984127 | 0.0001984127 × x^7 | 1 |
| 8 | 0.0000248016 | 0.0000248016 × x^8 | 1 |
| 9 | 0.0000027557 | 0.0000027557 × x^9 | 1 |
| 10 | 2.755732e-7 | 2.755732e-7 × x^10 | 1 |
FAQs
1) What does this calculator actually compute?
It differentiates a truncated Taylor series using the coefficients you provide. It returns the derived polynomial, the derivative value at a chosen point, and the derivative at the expansion center.
2) Why are coefficients entered instead of a full symbolic function?
Taylor coefficients are enough to reconstruct the truncated local model. Once the coefficients are known, derivative values and derivative polynomials follow directly without symbolic differentiation.
3) What is the meaning of the expansion center?
The center a is the point around which the series is built. Every term uses powers of (x-a), so changing a changes both the polynomial interpretation and derivative behavior.
4) Why can the derivative at the center be found so quickly?
At the center, every power of (x-a) becomes zero except the constant term of the derivative polynomial. That makes f^(k)(a) = k!c_k immediately available.
5) Why might the result differ from the exact derivative?
This tool uses a truncated series, not an infinite expansion. Accuracy depends on the number of terms, the evaluation point, and whether the point lies close to the expansion center.
6) Which preset functions are included?
The page includes presets for e^x, sin(x), cos(x), 1/(1-x), 1/(1+x), ln(1+x), and atan(x).
7) Can I use a nonzero center with custom coefficients?
Yes. The calculator is designed for any center value. Just make sure your coefficients correspond to powers of (x-a) around that same chosen center.
8) What do the CSV and PDF downloads contain?
They export the current calculation summary, including the chosen series, center, evaluation point, derivative order, numerical outputs, and the generated polynomial expressions.