Calculator Inputs
Formula Used
This calculator estimates the derivative from first principles. It shrinks h repeatedly and tracks how the difference quotient settles near a limit.
For each step, the tool computes a finite quotient with the current h value. The last quotient becomes the estimate, and a central-difference check helps confirm stability.
The trend label is based on the sign of the estimated derivative. Positive values suggest increasing behavior, negative values suggest decreasing behavior, and values near zero suggest a nearly flat tangent.
How to Use This Calculator
- Enter a function using x as the variable.
- Choose the point where you want the derivative.
- Set an initial h value that is reasonably small.
- Pick a shrink factor such as 0.1 or 0.5.
- Select the number of iterations for the quotient table.
- Enter a tolerance to judge convergence quality.
- Press the calculate button to show the result above the form.
- Review the quotient table, tangent line, and export buttons.
Example Data Table
Example function: f(x) = x2 + 3x at x = 2. The exact derivative is 2x + 3, so the expected value is 7.
| Step | h | f(2+h) | f(2) | Difference quotient |
|---|---|---|---|---|
| 1 | 1.0000 | 18.0000 | 10.0000 | 8.0000 |
| 2 | 0.5000 | 13.7500 | 10.0000 | 7.5000 |
| 3 | 0.1000 | 10.7100 | 10.0000 | 7.1000 |
| 4 | 0.0100 | 10.0701 | 10.0000 | 7.0100 |
FAQs
1. What does derivative by definition mean?
It means estimating the derivative from the limit of the difference quotient. Instead of using a memorized rule, you observe how secant slopes behave as h approaches zero.
2. Why does the calculator use several h values?
Several h values show whether the quotient is converging. A stable sequence gives more confidence that the last approximation is close to the true derivative.
3. What is a good starting h value?
A moderate small value, such as 1, 0.5, or 0.1, often works well. Extremely tiny values may create roundoff issues before the estimate stabilizes.
4. What does the central-difference check do?
It provides a secondary numerical estimate using points on both sides of x. If it closely matches the main estimate, your result is usually more trustworthy.
5. Can I enter trigonometric or logarithmic functions?
Yes. The calculator supports common functions such as sin, cos, tan, ln, log, log10, exp, sqrt, and abs, along with constants like pi and e.
6. Why might the result fail to converge?
The function may be undefined near the chosen point, the step settings may be poor, or floating-point precision may interfere with very small h values.
7. What does the tangent line output mean?
It uses the estimated derivative as the slope at your selected point. The displayed line is the local linear approximation of the function near that point.
8. What is included in the CSV and PDF downloads?
The exports include the input settings, summary results, and the full quotient table. That makes it easier to document classroom work or share calculations.