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.
coeff = 1n = 2x0 = 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
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