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

title
Equation of Tangent Line

Equation of Tangent Line

A tangent line to a curve is a straight line that touches a curve, or a graph of a function, at only a single point. The tangent line represents the instantaneous rate of change of the function at that one point. The slope of the tangent line at a point on the function is equal to the derivative of the function at the same point.

Finding Equation of the tangent line:

To find the equation of tangent line to a curve at point x=x0, we need to find the following:

1. Find the derivative of the function (i.e.derivative of the equation of curve).
2. Find the value of the derivative by putting x=x0 , this will be the slope of the tangent (say m).
3. Find the value y0, by putting the value of x0 in the equation of the curve. Our tangent will pass through this point (x0,y0)
4. Find the equation of the tangent using point-slope form. As the tangent passes through (x0,y0) and have slope m, the equation of the tangent line can be given as: (y-y0)=m.(x-x0)

Example : To find the equation of tangent line to the curve f(x) = 4x^2-4x+1 at x=1

Solution: f(x) = 4x^2-4x+1

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

Step 2 : m = f'(2) = 8.2-4 = 12

Step 3 : y0= f(x0) = f(2) = 4.2^2-4.2+1 = 16-8+1 = 9

Step 4 : m=12 ; (x0,y0)=(2,9)

Therefore, equation of tangent line is : (y-y0)=m.(x-x0)

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

=> y=12x-15