This is a continuation of Black-Scholes-Merton Derivation (Part 1: Portfolio Value Evolution).
Let 
According to the Ito-Doeblin formula, the differential of 
![\displaystyle\begin{aligned}dc(t,S(t))&=c_t(t,S(t))\, dt+c_x(t,S(t))\, dS(t)+\frac{1}{2}c_{xx}(t,S(t))\, dS(t)\, dS(t)\\&=c_t(t,S(t))\, dt+c_x(t,S(t))(\alpha S(t)\, dt+\sigma S(t)\, dW(t))+\frac{1}{2}c_{xx}(t,S(t))\sigma^2 S^2(t)\, dt\\&=\left[ c_t(t,S(t))+\alpha S(t) c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\right]\, dt\\&\quad+\sigma S(t)c_x(t,S(t))\, dW(t)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Ddc%28t%2CS%28t%29%29%26%3Dc_t%28t%2CS%28t%29%29%5C%2C+dt%2Bc_x%28t%2CS%28t%29%29%5C%2C+dS%28t%29%2B%5Cfrac%7B1%7D%7B2%7Dc_%7Bxx%7D%28t%2CS%28t%29%29%5C%2C+dS%28t%29%5C%2C+dS%28t%29%5C%5C%26%3Dc_t%28t%2CS%28t%29%29%5C%2C+dt%2Bc_x%28t%2CS%28t%29%29%28%5Calpha+S%28t%29%5C%2C+dt%2B%5Csigma+S%28t%29%5C%2C+dW%28t%29%29%2B%5Cfrac%7B1%7D%7B2%7Dc_%7Bxx%7D%28t%2CS%28t%29%29%5Csigma%5E2+S%5E2%28t%29%5C%2C+dt%5C%5C%26%3D%5Cleft%5B+c_t%28t%2CS%28t%29%29%2B%5Calpha+S%28t%29+c_x%28t%2CS%28t%29%29%2B%5Cfrac%7B1%7D%7B2%7D%5Csigma%5E2+S%5E2%28t%29c_%7Bxx%7D%28t%2CS%28t%29%29%5Cright%5D%5C%2C+dt%5C%5C%26%5Cquad%2B%5Csigma+S%28t%29c_x%28t%2CS%28t%29%29%5C%2C+dW%28t%29%5Cend%7Baligned%7D&bg=ffffff&fg=000000&s=0 "\displaystyle\begin{aligned}dc(t,S(t))&=c_t(t,S(t))\, dt+c_x(t,S(t))\, dS(t)+\frac{1}{2}c_{xx}(t,S(t))\, dS(t)\, dS(t)\\&=c_t(t,S(t))\, dt+c_x(t,S(t))(\alpha S(t)\, dt+\sigma S(t)\, dW(t))+\frac{1}{2}c_{xx}(t,S(t))\sigma^2 S^2(t)\, dt\\&=\left[ c_t(t,S(t))+\alpha S(t) c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\right]\, dt\\&\quad+\sigma S(t)c_x(t,S(t))\, dW(t)\end{aligned}")
The second equality uses the definition of the stock modeled by geometric Brownian motion 
Next, we compute the differential of the discounted option price 
![\displaystyle\begin{aligned}d(e^{-rt}c(t,S(t)))&=df(t,c(t,S(t)))\\&=f_t(t,c(t,S(t)))\, dt+f_x(t,c(t,S(t)))\, dc(t,S(t))\\&\quad+\frac{1}{2}f_{xx}(t,c(t,S(t)))\, dc(t,S(t))\, dc(t,S(t))\\&=-re^{-rt}c(t,S(t))\,dt+e^{-rt}\, dc(t,S(t))\\&=e^{-rt}[-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))\\&\quad+\frac{1}{2}\sigma^2S^2(t)c_{xx}(t,S(t))]\, dt+e^{-rt}\sigma S(t)c_x(t,S(t))\, dW(t).\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Dd%28e%5E%7B-rt%7Dc%28t%2CS%28t%29%29%29%26%3Ddf%28t%2Cc%28t%2CS%28t%29%29%29%5C%5C%26%3Df_t%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dt%2Bf_x%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%5Cquad%2B%5Cfrac%7B1%7D%7B2%7Df_%7Bxx%7D%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%3D-re%5E%7B-rt%7Dc%28t%2CS%28t%29%29%5C%2Cdt%2Be%5E%7B-rt%7D%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%3De%5E%7B-rt%7D%5B-rc%28t%2CS%28t%29%29%2Bc_t%28t%2CS%28t%29%29%2B%5Calpha+S%28t%29c_x%28t%2CS%28t%29%29%5C%5C%26%5Cquad%2B%5Cfrac%7B1%7D%7B2%7D%5Csigma%5E2S%5E2%28t%29c_%7Bxx%7D%28t%2CS%28t%29%29%5D%5C%2C+dt%2Be%5E%7B-rt%7D%5Csigma+S%28t%29c_x%28t%2CS%28t%29%29%5C%2C+dW%28t%29.%5Cend%7Baligned%7D&bg=ffffff&fg=000000&s=0 "\displaystyle\begin{aligned}d(e^{-rt}c(t,S(t)))&=df(t,c(t,S(t)))\\&=f_t(t,c(t,S(t)))\, dt+f_x(t,c(t,S(t)))\, dc(t,S(t))\\&\quad+\frac{1}{2}f_{xx}(t,c(t,S(t)))\, dc(t,S(t))\, dc(t,S(t))\\&=-re^{-rt}c(t,S(t))\,dt+e^{-rt}\, dc(t,S(t))\\&=e^{-rt}[-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))\\&\quad+\frac{1}{2}\sigma^2S^2(t)c_{xx}(t,S(t))]\, dt+e^{-rt}\sigma S(t)c_x(t,S(t))\, dW(t).\end{aligned}")
The third equality is due to 
Reference
Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.
Continued at: https://blog.nus.edu.sg/wuchengyuan/2022/08/13/black-scholes-merton-derivation-part-3-equating-portfolio-value-option-value-evolutions/