Equation of Line Tangent to Function Calculator

Compute tangent equations from entered functions confidently. Review slope, intercept, derivative estimate, and plotted curve. Save neat outputs for study, checking, reporting, and reuse.

Calculator input

Function f(x)

Point x₀

Derivative step h

Graph minimum x

Graph maximum x

Graph sample count

Optional minimum y

Optional maximum y

Decimal places

Input notes

Use x as the variable.
Use explicit multiplication such as 2*x.
Supported functions: sin, cos, tan, asin, acos, atan, sqrt, abs, exp, log, ln.
Use pi and e for constants.
log means base 10. ln means natural logarithm.

Example data table

Function	x₀	f(x₀)	Slope	Tangent line
x^2 + 3*x + 1	2	11	7	y = 7x - 3
sin(x)	0	0	1	y = x
x^3 - 2*x + 1	2	5	10	y = 10x - 15

Formula used

This calculator estimates the derivative numerically with the central difference method:

m ≈ [f(x₀ + h) - f(x₀ - h)] / (2h)

After finding the slope, it builds the tangent line through the point (x₀, f(x₀)):

y - f(x₀) = m(x - x₀)

It also calculates the slope-intercept form using b = f(x₀) - m x₀.

How to use this calculator

Enter the function using x as the variable.
Choose the tangent point x₀.
Set a small derivative step h, such as 0.0001.
Choose the graph range and sample count.
Press calculate to see equations, values, graph, and download options.

Frequently asked questions

1. What is a tangent line?

A tangent line touches a curve at one chosen point and shares the same instantaneous slope there. It gives the best local linear description near that point.

2. What does the slope mean here?

The slope is the derivative value at x₀. It shows how fast the function is changing at that exact point and determines the tilt of the tangent line.

3. Why does the calculator use h?

The value h sets the tiny step used in the central difference estimate. A small h usually improves the slope estimate, but an extremely small value may increase rounding noise.

4. Can I use trigonometric functions?

Yes. You can use sin, cos, tan, asin, acos, and atan. Angles are interpreted in radians, which matches standard calculus and programming conventions.

5. What if the function is undefined at the point?

If the function or its nearby values are undefined, the calculator will show an error. A tangent line needs a valid point and a usable local slope estimate.

6. Why does the graph sometimes have gaps?

Gaps appear when the function is undefined for some x-values inside the selected range. Common examples include division by zero, square roots of negatives, or invalid logarithms.

7. What forms of the tangent line are shown?

The result includes point-slope form, slope-intercept form, standard form, and a linear approximation. These forms help with checking, teaching, and further algebra steps.

8. What gets exported in CSV and PDF?

The exports include the main tangent line results. The CSV also includes graph sample data, while the PDF captures the summary values and the plotted graph.