Ito product rule

Stochastic Differential Equation
$$dS(t) = rS(t)(α − S(t))dt + σS(t)dW(t), S(0) = x.$$
where W(t) is a standard Brownian motion and r, α, σ are positive constants. I have no idea why this question need ito product rule. Show by using the product rule that the following process is a closed-form solution of the above Stochastic Differential Equation

$$S(t)=\frac{\exp\big\{(r\alpha-\frac{1}{2}\sigma^2)t+\sigma W(t)\big\}}{\frac{1}{x}+r\int_0^t\exp\big\{(r\alpha-\frac{1}{2}\sigma^2)s+\sigma W(s)\big\}\ ds}$$

I have no idea why this question need ito product rule.

My thought is let $$f(Z)=\frac{Z}{(1/x+r\int z\ ds)}$$
$$dS(t)=df(z(t))=fz(z(t))dz(t)+fzz(z(t))dz(t)dz(t)$$, but I'm stuck.