Using Newton's Method to Solve Transcendental Equations
Newton's method is most useful
when applied to complicated functions where we have no way to get an
algebraic solution, even with the help of a computer.
For instance, solving $e^x + x = 4$ is the
same as finding the roots of $f(x) = e^x+x-4$.
As always, we have $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)},$$
which in this case becomes
$$x_n - \frac{e^{x_n}+x_n-4}{e^{x_n}+1}.$$