I want to know how the derivative of $C(S,t)$ with respect to t is calculated (theta). A lot of sources just mention "with a bit of manipulation.." but I have been unsuccessful in understanding this manipulation.
Similarly, I am trying to understand how $\frac{\partial^{2} C }{\partial S^{2}} = \frac{N(d1)}{S\sigma\sqrt{T-t}}$
From the formula, $$ d_1= \frac{ \ln(S/K) + (r-\delta+\frac12 \sigma^2)(T-t)}{\sigma \sqrt{T-t}}, $$ we have $$ \frac{\partial d_1}{\partial S} = \frac1{S\sigma\sqrt{T-t}}.$$ This is because $S$ is only included in $\ln(S/K)$ and the derivative of it with respect to $S$ is $1/S$. Also, note that there is $\sigma \sqrt{T-t}$ term on the denominator.