The first derivative comes from the squared-error loss. From there, the output layer and hidden activations receive exact gradients by multiplication.
Backward from the loss
Squared error gives dL/dyhat=-2 because yhat=2 and y=3.
dy^dL=2(y^−y)=2(2−3)=−2
Output-layer parameter gradients
The output weights multiply the hidden activations. Since h1=2 and h2=0, the exact gradients are dL/dv1=-4, dL/dv2=0, and dL/dc=-2.
dv1dL=−2⋅2=−4,dv2dL=−2⋅0=0
Back into the hidden activations
The same output gradient flows through v1 and v2. Here dL/dh1=-2 and dL/dh2=-2.
dh1dL=−2,dh2dL=−2