Calculate tangent slopes from several curve models quickly. Get normal lines, derivatives, and rate checks. Export results, inspect graphs, and learn each worked step.
| Mode | Curve or points | x value | Expected slope |
|---|---|---|---|
| Polynomial | y = x² + 3x + 2 | 2 | 7 |
| Exponential | y = 2e^(0.5x) | 0 | 1 |
| Two points | (1,2) and (3,10) | midpoint | 4 |
The slope of a curve shows how fast y changes as x changes. A straight line has one fixed slope. A curve usually changes slope from point to point. That is why tangent slope matters. It gives the local rate of change at one chosen x value.
The derivative gives the exact tangent slope when a formula is known. A secant slope compares two nearby points. It gives an average rate of change over a short interval. When the interval becomes very small, the secant slope approaches the tangent slope. This calculator shows both values so you can compare exact and approximate results.
This page does more than a simple first derivative check. It returns the curve value, tangent slope, numeric slope, nearby secant slope, tangent line, normal line, second derivative, curvature, and radius of curvature when possible. These extra values help with calculus homework, graph interpretation, optimization studies, and engineering style rate analysis.
Several common models are included. You can test polynomial, power, exponential, logarithmic, trigonometric, and rational curves. A separate two point mode is also included. That option is useful when no explicit formula is available and only measured coordinates are known.
The graph helps you see whether the tangent line matches the local shape of the curve. A positive slope rises to the right. A negative slope falls to the right. A zero slope is horizontal. A steep slope means rapid change. The visual view makes these ideas easier to understand.
Some inputs have domain limits. Logarithmic curves need a positive inside term. Rational curves fail where the denominator becomes zero. Tangent functions can break near vertical asymptotes. If a value is not valid, the calculator reports that clearly so you can adjust the input and try again.
The slope of a curve is the rate of change at a chosen point. In calculus, it is the derivative value at that x position.
Tangent slope is local and exact at one point. Secant slope is average change between two points. A very small interval makes them close.
The normal line is perpendicular to the tangent line. It is useful in geometry, optics, curve analysis, and many applied maths problems.
N/A appears when the selected inputs break the curve domain or make the slope undefined. Common cases include log limits and zero denominators.
Step size h controls the central difference estimate. Smaller values usually improve local slope estimates, but extremely small values can add rounding noise.
Yes. Choose the two point secant mode. It calculates average slope, midpoint, secant line, and a normal line at the midpoint.
Curvature measures how sharply a curve bends at a point. Larger curvature means tighter bending. The radius of curvature is its reciprocal.
Pick the mode that matches your equation form. If your data comes from observations instead of a formula, start with the two point option.
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.