About

The tangent line to a curve at a given point is the straight line that "just touches" the curve at that point. In calculus, the slope of this line is determined by the derivative of the function, denoted as f'(x).

This tool automates the process: it takes a polynomial function f(x) and an x-coordinate, computes the derivative to find the slope m, calculates the y-coordinate, and uses the point-slope form to generate the final linear equation y = mx + b. It visualizes the intersection on a canvas graph.

Formulas

1. Find Derivative: Compute f'(x) using the Power Rule:

d/dx (xⁿ) = nx^n-1

2. Find Slope: Evaluate m = f'(x₀).

3. Point-Slope Form: Use y₀ = f(x₀):

y − y₀ = m(x − x₀)

Reference Data

Function f(x)	Point x	Derivative f'(x)	Slope (m)	Tangent Equation
x²	1	2x	2	y = 2x − 1
x³	1	3x²	3	y = 3x − 2
2x² + x	0	4x + 1	1	y = x
x² − 4	2	2x	4	y = 4x − 8
x⁴	-1	4x³	-4	y = -4x − 3
5	3	0	0	y = 5
x	5	1	1	y = x
-x²	2	-2x	-4	y = -4x + 4

Frequently Asked Questions

The Power Rule is a basic technique in calculus for finding the derivative of a variable raised to a power. It states that the derivative of x^n is nx^(n-1).

The tangent line represents the instantaneous rate of change of the function at that specific point. It is fundamental to approximation methods in physics and engineering.

This simplified tool currently processes polynomial functions (e.g., x^2 + 2x). It does not parse trigonometric or logarithmic functions.

A slope of 0 indicates a horizontal tangent line. This usually occurs at local maximums or minimums (peaks or valleys) of the curve.