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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)}.