Calculator Input
Example Data Table
This example uses f(x) = 3*x^2 + 2*x + 1 at x = 2 with a small step size. The exact derivative is 14.
| x | f(x) |
|---|---|
| 1.98 | 16.7212 |
| 1.99 | 16.8603 |
| 2.00 | 17.0000 |
| 2.01 | 17.1403 |
| 2.02 | 17.2812 |
Formula Used
The instantaneous rate is the derivative at a chosen point.
f'(a) = lim h→0 [f(a+h) - f(a)] / h
This calculator estimates that derivative numerically with selectable finite-difference methods.
Central difference: f'(a) ≈ [f(a+h) - f(a-h)] / (2h)
Five-point stencil: f'(a) ≈ [-f(a+2h) + 8f(a+h) - 8f(a-h) + f(a-2h)] / (12h)
Approximate second derivative: f''(a) ≈ [f(a+h) - 2f(a) + f(a-h)] / h^2
How to Use This Calculator
Enter a function in terms of x. Use explicit multiplication, such as 4*x or 2*sin(x).
Choose the point where you want the instantaneous rate. Enter a small positive step size h.
Select a derivative method. Five-point stencil is usually the most accurate option for smooth functions.
Set the decimal precision and optional unit labels. Submit the form to view the rate, tangent line, angle, and nearby values.
Use the export buttons after calculation to save a CSV summary or a PDF report.
FAQs
1. What does instantaneous rate mean?
It is the slope of the function at one exact point. In calculus, that slope is the derivative and describes how fast the output changes right there.
2. Why is a step size needed?
The calculator estimates the limit process numerically. A small step size samples nearby points so the derivative can be approximated with finite-difference formulas.
3. Which method should I choose?
For smooth functions, the five-point stencil usually gives the best balance of accuracy and stability. Central difference is also strong and slightly simpler.
4. Can I use trigonometric and logarithmic functions?
Yes. You can use functions like sin, cos, tan, sqrt, abs, exp, ln, and log. Write them in standard function format with parentheses.
5. Why do I need explicit multiplication?
Expressions are safer and clearer when multiplication is written directly. Use 3*x^2 instead of 3x^2, and 2*sin(x) instead of 2sin(x).
6. What if the function is undefined near the point?
The calculator will show an error. Derivative estimates need valid nearby values, so discontinuities, zero denominators, or invalid logs can block computation.
7. What does the tangent line show?
The tangent line touches the curve at the selected point and shares the same slope there. It is a local linear model for the function.
8. Why export CSV or PDF?
Exports help with homework, reports, and checking results later. CSV works well for spreadsheets, while PDF is useful for printing or sharing.