Final expression

Defining

$$ \delta_1^{(1)}=w_1^{(2)}\frac{\partial a_1^{(1)}}{\partial z_1^{(1)}}\delta^{(2)}, $$

we have

$$ \frac{\partial C}{\partial w_{11}^{(1)}}=\delta_1^{(1)}a_1^{(1)}. $$

Similarly, we obtain

$$ \frac{\partial C}{\partial w_{12}^{(1)}}=\delta_1^{(1)}a_2^{(1)}. $$