Calculator
Use the responsive grid below for the function, point, and graph settings.
Plotly graph
The chart below compares the original function, the tangent line, and the tangency point.
Example data table
| Function family | Function | x₀ | Point | Slope | Tangent line |
|---|---|---|---|---|---|
| Polynomial | y = x² + 3x + 1 | 2 | (2, 11) | 7 | y = 7x - 3 |
| Exponential | y = 2e^x | 0 | (0, 2) | 2 | y = 2x + 2 |
| Logarithmic | y = ln(x) | 1 | (1, 0) | 1 | y = x - 1 |
| Sine | y = sin(x) | 0 | (0, 0) | 1 | y = x |
Formula used
The calculator uses the tangent-line formula y - f(x₀) = f′(x₀)(x - x₀). It first evaluates the function value at the chosen point, then evaluates the derivative at the same point, and finally builds the tangent equation from those two quantities.
| Family | Function form | Derivative used |
|---|---|---|
| Polynomial | f(x) = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀ | f′(x) = 4a₄x³ + 3a₃x² + 2a₂x + a₁ |
| Power | f(x) = a(x - h)^n + c | f′(x) = an(x - h)^(n - 1) |
| Exponential | f(x) = ae^(b(x - h)) + c | f′(x) = abe^(b(x - h)) |
| Logarithmic | f(x) = a ln(b(x - h)) + c | f′(x) = ab / (b(x - h)) |
| Trig models | f(x) = a sin, cos, or tan of b(x - h) + c | Standard trig derivatives, with π/180 included for degree mode |
| Reciprocal | f(x) = a / (b(x - h)) + c | f′(x) = -(ab) / (b(x - h))² |
| Square root | f(x) = a√(b(x - h)) + c | f′(x) = ab / (2√(b(x - h))) |
How to use this calculator
- Select a function family that matches your equation style.
- Enter the coefficients, shifts, and exponent where needed.
- Set the tangency x-value, graph window, and sample count.
- Use degree mode only for sine, cosine, or tangent models.
- Click the calculate button to show the result above the form.
- Review the tangent equation, slope, intercepts, and chart.
- Download CSV for tabular output or PDF for a clean report.
FAQs
1. What does this calculator find?
It finds the tangent line to a selected function at a chosen x-value. It also shows the derivative, point of tangency, slope, intercepts, normal line, and a comparison graph.
2. What is a tangent line?
A tangent line touches a curve at one point and has the same instantaneous slope there. Near that point, the line gives a strong local linear approximation of the function.
3. Why do I need the derivative?
The derivative gives the slope of the curve at the selected point. Once the slope and the point coordinates are known, the tangent equation follows directly from point-slope form.
4. Can I use degree mode for all functions?
Degree mode only matters for sine, cosine, and tangent models. Polynomial, logarithmic, exponential, reciprocal, power, and square-root forms do not use angular units.
5. Why do some entries return an error?
Errors happen when the model is undefined at the chosen point, such as taking a logarithm of a non-positive value, dividing by zero, hitting a tangent asymptote, or differentiating a square root at a boundary point.
6. What is the normal line?
The normal line passes through the same point as the tangent line but is perpendicular to it. Its slope is the negative reciprocal of the tangent slope, when that reciprocal exists.
7. Why are exports useful?
CSV export helps with spreadsheets, assignments, and checking sample values. PDF export creates a compact summary you can save, share, or attach to notes and reports.
8. Does the graph prove the answer?
The graph is a visual check, not the proof itself. The exact result still comes from evaluating the function and derivative formulas at the selected tangency point.