Partial Derivative in Linear Model, how are they calculated?

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Here is a flowchart I am struggling with: enter image description here

I am a programmer with a weak math background. I don't understand for example how the partial derivative of $\hat{y}$ in respect to weight 1 equals $= x_1$.

I understand the importance of them, however I cannot see how to calculate them.I have been on Khan Academy and nothing is clicking any assistance would be much appreciated!

Here is the article that uses the chart for further context.