RNN-BPTT (ch9.7.1) partial derivative notation confusion

[kennethjang34]

Hello! I am a CS student and have been studying DL in his spare time with D2L textbook:)
I'd like to tell you that I really appreciate your work and have really loved the book! Thank you very much.

When I was reading RNN part, I am a bit confused about how we get the partial derivative of hidden state value with respect to w_h.

in 9.7.1 Analysis of Gradients in RNNs,
for formula 9.7.4, we have:
$$\frac{\partial h_t}{\partial w_h}= \frac{\partial f(x_{t},h_{t-1},w_h)}{\partial w_h} +\frac{\partial f(x_{t},h_{t-1},w_h)}{\partial h_{t-1}} \frac{\partial h_{t-1}}{\partial w_h}.$$

However, as we have the following from formula 9.7.1
$$h_t:=f(x_t,h_{t-1},w_h).$$,
isn't it
$$\frac{\partial h_t}{\partial w_h}-\frac{\partial f(x_{t},h_{t-1},w_h)}{\partial w_h}=0.$$ ?
I was wondering if
$$\frac{\partial h_t}{\partial w_h}, \frac{\partial h_{t-1}}{\partial w_h}.$$
should be ordinary deriviates of some function representing h_t that can be parameterized with w_h.