The Newton Raphson method does not need a change of sign, but instead uses the tangent to the graph at a known point to provide a better estimate for the root of the equation.

Here our new estimate for the root is found using the iteration:

Note: f'(x) is the differential of the function f(x).

(This equation is essentially saying you must divide the y-value by the gradient, and subtract this from the previous estimate.)

Repeat the process until the root is found to the desired degree of accuracy.

Example:

Lets take the equation f(x) = e^x − 3x.

Differentiating this gives f'(x) = e^x - 3.

Let's estimate that the root to this equation is 0.6.

Then a better estimation of the answer will be given by the Newton Raphson equation:

We can take this better estimation and put it through the Newton Raphson equation again to get an even more accurate solution:

As this gives the same solution, it must be the correct root for the equation (to 3 decimal places).

Note: If the gradient = 0 at the starting point then no solution can be found by this method (because we cannot divide by 0).