Enter a function and point to differentiate fast. Pick method, step, and view tangent details. Download clean reports, compare slopes, and learn better today.
Sample values for f(x)=x² at x₀=2 with h=0.1 (for demonstration).
| x | f(x)=x² | Approx slope |
|---|---|---|
| 1.8 | 3.24 | ≈3.60 |
| 1.9 | 3.61 | ≈3.80 |
| 2.0 | 4.00 | ≈4.00 |
| 2.1 | 4.41 | ≈4.20 |
| 2.2 | 4.84 | ≈4.40 |
If your function has a limited domain, use one-sided methods.
In calculus, slope represents instantaneous change, which powers optimization and sensitivity testing. When you evaluate f′(x₀), you measure how much the output shifts for a tiny input shift at x₀. Engineers use slope to approximate load–deflection response, while analysts use it to estimate marginal effects in cost and demand curves.
The step h controls the balance between truncation error and rounding error. Large h smooths noise but can ignore curvature, while extremely small h can magnify floating‑point limits. A practical workflow is to start at 0.001, then test 0.002 and 0.0005; if the slope stabilizes across nearby h values, the estimate is reliable for your function near x₀.
Central difference uses values on both sides of x₀ and typically achieves better accuracy for smooth functions because the first‑order error terms cancel. Forward and backward differences are valuable at boundaries, such as sqrt(x) near zero or when your domain restricts x−h. Comparing methods can also flag non‑smooth behavior, where slopes differ dramatically across directions.
The tangent line is a local linear model: y = m(x − x₀) + f(x₀). It is most useful for short moves around x₀, where curvature is modest. The intercept form shown by the calculator makes it easy to plug in new x values. This supports quick decisions. If the second‑derivative estimate is large, keep your tangent predictions within a tighter neighborhood.
People often understand slope better as an angle or percent grade. The calculator converts m into θ = arctan(m) and grade = 100m. For example, m = 0.10 corresponds to roughly 5.71° and a 10% grade. Negative slopes produce negative angles, matching downhill direction in geometric and physical interpretations.
The sample table around x₀ lists nearby function values and local slopes. Consistent slopes across the row suggest a stable derivative region, while rapid changes hint at strong curvature or a cusp. For classroom checks, try polynomials where the true derivative is known, then confirm the numeric slope approaches the analytic value as h is refined.
It estimates f′(x₀) using a finite‑difference method. The result approximates the tangent slope at the chosen point, along with the tangent line equation and a readable angle and grade.
Central difference is usually best for smooth functions because it balances both sides of x₀. Use forward or backward differences when the function is undefined on one side of the point.
Start with 0.001, then try nearby values like 0.0005 and 0.002. If the slope stays similar, your estimate is stable. If it jumps, increase h or review domain issues.
The function must be defined at x₀ and at the nearby points used by your method, such as x₀±h. Choose an x₀ inside the domain or switch to a one‑sided method.
The angle is θ = arctan(m), where m is the slope. It describes the tilt of the tangent line relative to the positive x‑axis, expressed in degrees and radians.
Exports include your function, x₀, f(x₀), the selected method, h, the computed slope, tangent line, and a small table of nearby points with local slope estimates for quick validation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.