How does the Jacobian matrix help extend Newton’s method to systems of nonlinear equations?

Newton’s method for a single nonlinear equation $f(x)=0$ uses the derivative $f^{\prime }(x)$ to approximate the function with a tangent line, leading to the iteration $x^{(k+1)}=x^{(k)}-f(x^{(k)})/f^{\prime }(x^{(k)})$.1

To extend this to a system of $n$ nonlinear equations $𝐟(𝐱)=0$, the scalar derivative is replaced by the Jacobian matrix, denoted as ${𝐟}^{\prime }(𝐱)$. The Jacobian matrix is a square matrix consisting of the first partial derivatives of each function with respect to each variable.12

The generalized Newton’s method for systems is defined by the iteration:
${𝐱}^{(k+1)}={𝐱}^{(k)}+{\mathbf{\delta }}^{(k)}$
where ${\mathbf{\delta }}^{(k)}$ is found by solving the linear system:
${𝐟}^{\prime }({𝐱}^{(k)}){\mathbf{\delta }}^{(k)}=-𝐟({𝐱}^{(k)})$1

In this approach, the Jacobian matrix allows the method to linearize the system at each step, effectively reducing the nonlinear problem to a sequence of linear systems of order $n$ that can be solved to find successively better approximations of the root.1