proof of Newton-Girard formula for symmetric polynomials

The following is a proof of Newton-Girard formula using formalpower series. Let $z$ be an indeterminate and $f(z)$ be thepolynomial

$$1-E_{1}z+\mathrm{\dots }+{(-1)}^n{E}_n{z}^n.$$

Take log and differentiate both sides of the equation

$$f(z)=\prod _{i=1}^n(1-x_{i}z).$$

We obtain

$$f^{\prime }(z)/f(z)=\sum _{i=1}^n\frac{-x_i}{1-x_{i}z},$$(1)

where $f^{\prime }(z)$ is the derivative of $f(z)$

$$f^{\prime }(z)=-E_1+2E_{2}z-\mathrm{\dots }+{(-1)}^{n}n{E}_n{z}^{n-1}.$$

The right hand side of (1) is equal to

$$-\sum _{i=1}^n\sum _{k=0}^{\mathrm{\infty }}{x}_i^{k+1}{z}^k=-\sum _{k=0}^{\mathrm{\infty }}{S}_{k+1}{z}^k.$$

By equating coefficients of

$$f^{\prime }(z)=-f(z)(S_1+S_{2}z+S_3{z}^2+\mathrm{\dots })$$

we get the Newton-Girard formula.