Slope of Tangent Line Calculator Formula
Understand the math behind the slope of tangent line calculator. Each variable explained with a worked example.
Formulas Used
Y0
y0 = coeff * pow(x0, n)Slope (f'(x₀))
slope = coeff * n * pow(x0, n - 1)Y Intercept
y_intercept = coeff * pow(x0, n) - coeff * n * pow(x0, n - 1) * x0Variables
| Variable | Description | Default |
|---|---|---|
coeff | Coefficient (a) | 1 |
n | Exponent (n) | 2 |
x0 | Point x₀ | 3 |
How It Works
Slope of a Tangent Line
Method
For f(x) = ax^n, the tangent line at x₀ has:
1. Slope = f'(x₀) = a × n × x₀^(n-1) 2. Point = (x₀, f(x₀)) 3. Tangent line equation: y - y₀ = m(x - x₀)
or equivalently: y = mx + (y₀ - mx₀)
Worked Example
Find the tangent line to y = x² at x = 3.
- 01f(3) = 3² = 9, so the point is (3, 9)
- 02f'(x) = 2x, so f'(3) = 6
- 03Tangent line: y - 9 = 6(x - 3)
- 04y = 6x - 9
Frequently Asked Questions
What is a tangent line?
A tangent line touches a curve at exactly one point and has the same slope as the curve at that point. It represents the best linear approximation of the function near that point.
How is the tangent slope related to the derivative?
The slope of the tangent line at a point is exactly the value of the derivative at that point. The derivative gives the instantaneous rate of change.
What is the normal line?
The normal line is perpendicular to the tangent line. If the tangent slope is m, the normal slope is -1/m.
Learn More
Guide
Introduction to Derivatives - Complete Guide
Learn the fundamentals of derivatives in calculus. Covers the definition, power rule, product rule, chain rule, and practical applications with clear examples.
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