Newton-Raphson Method (link)

A graphical representation can also be very helpful. Below, you see the same function f(x) = x2-4 (shown in blue). The process here is the same as above. In the first iteration, the red line is tangent to the curve at x0. The slope of the tangent is the derivative at the point of tangency, and for the first iteration is equal to 12. Dividing the value of the function at the initial x (f(6)=32) by the slope of the tangent (12), we find that the delta-x is equal to 2.67. Subtracting this from six (6) we find that the new x-value is equal to 3.33. Another way of considering this is to find the root of this tangent line. The new x-value (xn+1) will be equal to the root of the tangent to the function at the current x-value (xn).