What Each Box Hands Back

In a backward pass, no box ever looks at the whole graph. Each box receives one number from above — the gradient flowing back, $\dfrac{\partial L}{\partial (\text{its output})}$ — and multiplies it by its own local derivatives to hand one number back to each of its inputs. That is the chain rule, done one box at a time.

Task: write node_backward(op, inputs, upstream), which plays the part of a single box.

• op names the box.
• inputs lists the numbers that flowed into the box during the forward pass.
• upstream is the gradient arriving at the box's output from above.

Return a list holding one gradient for each input value, in the same order as inputs, each rounded to 4 decimal places.

For "pow", the exponent k is a fixed number written into the box — it may be negative or a fraction — not a value flowing through the graph. It gets no gradient, so a "pow" box returns a one-element list. Whenever k is a fraction, x is positive.

$\tanh$ is the one box this course has not differentiated, so take its local derivative as given: if $t = \tanh(x)$, then