Math

Tangent Line Calculator

Find the Equation of the Tangent Line

The tangent line touches a curve at exactly one point and has the same slope as the curve at that point. This calculator finds the derivative (slope), evaluates it at your chosen point, and gives the tangent line equation in point-slope form: y = f'(x0)(x - x0) + f(x0).

Derivative Calculator Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Power Rule

Differentiate a polynomial using the power rule.

Key values: f(x) = x^3 - 2x^2 + x · 1st derivative

Chain Rule

Differentiate a composite function with sin(x^2).

Key values: f(x) = sin(x^2) · Chain rule applied

Product Rule with Tangent

Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.

Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0

Second Derivative

Compute the second derivative to analyze concavity.

Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative

Documentation

This calculator is also known as Tangent Line Calculator.

Read the complete guide

What Is a Tangent Line?

A tangent line is a straight line that touches a curve at a single point and has the same slope as the curve at that point. The slope of the tangent line equals the derivative of the function at the point of tangency.

Tangent Line Formula

The tangent line equation uses the point-slope form:

Find the slope: m = f'(x0)
Find the point: (x0, f(x0))
Write the equation: y = m(x - x0) + f(x0)

Examples

Tangent to a Parabola

Find the tangent line to y = x^2 at x = 3.

f'(x) = 2x, so slope at x = 3 is 6. f(3) = 9. Tangent: y = 6(x - 3) + 9 = 6x - 9.

Key takeaway: The tangent line approximates the curve near the point of tangency.

Working with Tangent Lines

Key practices for tangent line problems:

Always find the derivative first -- the slope is the most important piece
Double-check by plugging x0 into both f(x) and f'(x)
Use the graph to verify that your tangent line looks correct

Frequently Asked Questions about Tangent Line Calculator

How do I find the equation of a tangent line?

Three steps: (1) Differentiate the function to get f'(x). (2) Evaluate f'(x0) for the slope m. (3) Use point-slope form: y = m(x - x0) + f(x0).

What is the difference between tangent and secant lines?

A tangent line touches the curve at exactly one point and has the same slope as the curve there. A secant line intersects the curve at two points. As the two points get closer together, the secant line approaches the tangent line.