Newton-Raphson's theorem, commonly known as **Newton-Raphson method**, is an iterative numerical method used to find the approximate roots (solutions) of a real-valued function \( f(x) \). It's particularly useful when finding exact roots is challenging. The method is based on linear approximation (tangent line) of the function and refines guesses to converge towards an accurate solution.

### Formula:
Given a differentiable function \( f(x) \), the Newton-Raphson iteration formula is:

\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]

Where:
- \( x_n \) is the current approximation of the root.
- \( f'(x_n) \) is the derivative of \( f(x) \) at \( x_n \).
- \( x_{n+1} \) is the next approximation.

### Steps of the Newton-Raphson Method:
1. Start with an initial guess \( x_0 \).
2. Compute the next approximation using the formula.
3. Repeat the process until the difference between successive approximations is sufficiently small, indicating convergence to a root.

### Example:
Suppose we want to find the root of the equation \( f(x) = x^2 - 2 \) (the square root of 2).

1. Let the initial guess \( x_0 = 1 \).
2. Calculate the next approximation using the formula:

\[
x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 1 - \frac{1^2 - 2}{2 \cdot 1} = 1 - \frac{-1}{2} = 1.5
\]

3. Repeat the process until \( x_n \) converges to a solution.

This method converges quickly if the initial guess is close to the actual root.

### Key Points:
- The method works best when the function is well-behaved and differentiable near the root.
- If the initial guess is far from the root, the method may fail to converge or converge to a wrong root.
- It is widely used in calculus, numerical analysis, and solving engineering problems.

Would you like more details or have any questions?

### Related Questions:
1. How does the Newton-Raphson method compare to other root-finding methods like bisection or secant method?
2. What are the conditions required for the Newton-Raphson method to converge?
3. How can the method fail, and what strategies can be used to avoid failure?
4. What are some real-world applications of the Newton-Raphson method?
5. How can Newton-Raphson be extended to systems of nonlinear equations?

**Tip**: Always check the derivative \( f'(x) \) doesn't vanish, as division by zero would cause failure of the method.

Explore the Newton-Raphson method, an iterative approach to approximate the roots of differentiable functions. This method leverages calculus to refine guesses towards accurate solutions. Learn the formula, step-by-step process, and examples for real-world applications.

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