The Bisection method for solving equations is accurate but quite lengthy. A quicker method, which requires differentiation, is the Newton Rhapson method.
In this method, an initial value of x, which is thought to be near the root is found. This value becomes x_{0}.
A tangent to the curve is drawn at this point and extended to cut the xaxis. This value is x_{1}.
This value is found using the general formula :
Note that the function f(x) must be differentiable.
When the equation has more than one root, the initial value of x must be near to the root being found or the second value of x could be nearer to another root of the equation.
This process is repeated until successive differences are less than the degree of accuracy required. i.e. I x_{ n } x_{n1}I < E
Example
Use the NewtonRhapson method to solve the equation x^{3} = 5 to 6 decimal places.
First of all, rearrange the equation to x^{3} − 5 = 0
Now, find an initial value.
x

0

1

2

3

x^{3 } 5

5

4

3

22

The root is between 1 and 2 (the sign changes from negative to positive) and nearer to 2, so we will use and initial value of x_{0} = 1.75
Differentiate f(x) = x^{3} − 5
f '(x) = 3x^{2}
Using
x_{1} = 1.75 − = 1.7108844
This process is now repeated until the required degree of accuracy is reached.
To summarise and continue this process in a table:
Initial value

x_{0}

1.75

x_{ n } x_{n1}

1st iteration

x_{1}

1.7108844

0.0391156

2nd iteration

x_{2}

1.7099764

0.000908

3rd iteration

x_{3}

1.7099759

0.0000005

As I x_{ n } x_{n1}I < 0.000001 the process is stopped.
The solution is x = 1.709976 (to 6 decimal places)
Nonconvergence
If the first value used in the Newtonrhapson method is not chosen correctly it could mean that successive iterations will not converge on the required root. An example is shown below .
The initial value x_{0} is too far from the required root and the tangents can be seen to be oscillating and not converging on the root.
Other cases of nonconvergence
If the initial value is a turning point, the tangent will not intersect the xaxis.
If the initial value is too far from the required root then convergence may be to another root.
Comparison of the Bisection Method and the Newton Rhapson Method
Comparing the two methods:
Method

Advantages

Disadvantages

Bisection 
Reliable Always converges to the root 
Slow, can require many iterations 
NewtonRhapson 
Fast convergence Simple formula 
Requires differentiation Can converge to wrong root May not converge at all 
A third method, the secant method can also be used.