freeCodeCamp/guide/english/mathematics/equation-of-tangent-line/index.md

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Equation of Tangent Line

Equation of Tangent Line

A tangent line to a function f(x) is a straight line that passes through the point (x₀, f(x₀)) and has slope f'(x₀). The slope of the tangent line represents the instantaneous rate of change of the function at that point.

Finding the equation of a tangent line:

To find the equation of a tangent line,

y = mx + b,

of a function f(x) at point x = x₀, we need to do the following:

1. Find the derivative of the function.
2. Find the value of the derivative at x = x₀, this will be the slope of the tangent (our m above).
3. Find the value y₀ of the function at x₀. The tangent will pass through the point (x₀, y₀).
4. Find the equation of the tangent using point-slope form. As the tangent passes through (x₀, y₀) and has slope m, the equation of the tangent line can be written as (y - y₀) = m(x - x₀) or y = mx + (y₀ - mx₀).

Example

Find the equation of tangent line to the function f(x) = 4x² - 4x + 1 at x = 2.

We proceed through the steps above.

Step 1 : f'(x) = 8x - 4.

Step 2 : m = f'(2) = 8 · 2 - 4 = 12.

Step 3 : y₀ = f(x₀) = f(2) = 4 · 2² - 4 · 2 + 1 = 16 - 8 + 1 = 9.

Step 4 : From steps 2 and 3 we have m = 12 and (x₀, y₀) = (2,9), so the equation of the tangent line is

(y - 9) = 12(x - 2)

or, rearranging to slope intercept form,

y = 12x - 15.