I'm very new to this forum. I'm learning the mathematics behind Neural Networks.
I have the error calculating formula which is the following:
$$\frac{\partial E_{\text{Total}}}{\partial \text{Out}_{y_1}}$$
The formula to calculate $E_{\text{total}}$ is:
$$1/2(T_{1} - \text{Out}_{y_1})^2 + 1/2(T_{2} - \text{Out}_{y_2})^2$$
The formula to calculate $\text{Out}_{y_1}$ is:
$$1/(1+e-^{y_1})$$
Now, I want to calculate the partial differentiation, which is $$\frac{\partial E_{\text{Total}}}{\partial \text{Out}_{y1}}~.$$ Could you please help me to understand the steps to solve this.
You gave the formula of
$$E_{\text{total}} = \left(\frac{1}{2}\right)(T_{1} - \text{Out}_{y_1})^2 + \left(\frac{1}{2}\right)(T_{2} - \text{Out}_{y_2})^2 \tag{1}\label{eq1}$$
Partial differentiation is similar to normal differentiation except that you treat the independent variables as constants when differentiating. Thus, assuming $T_1$, $T_2$ and $\text{Out}_{y_2}$ are independent of $\text{Out}_{y_1}$, using the differentiation Chain rule and Power rule gives
$$\begin{equation}\begin{aligned} \frac{\partial E_{\text{Total}}}{\partial \text{Out}_{y1}} & = \left(\frac{1}{2}\right)2(T_{1} - \text{Out}_{y_1})\frac{\partial (T_{1} - \text{Out}_{y_1})}{\partial \text{Out}_{y1}}\\ & = (T_{1} - \text{Out}_{y_1})(-1) \\ & = \text{Out}_{y_1} - T_1 \end{aligned}\end{equation}\tag{2}\label{eq2}$$