Hi,everyone,I got a problem here that I don't understand.Can anyone please help me?

The relation between three-dimentional cartesian coordinates and cylindrical coordinates is given by

Calculate the partial derivatives , and in terms of , , .

My current problem here is like a puzzle. I certainly can solve the question and find the answer. For example, I did find .Same as solution

However, my problem occured when I want to calculate ,if I just use , I would get as well!

I think it is weird, because I thought supposed to equal to the reciprocal of , which means it supposed to be .

So pretty sure is the right answer(same as solution),but why the recipocal rule does not apply here? I'm so confused about this puzzle. Same as two other questions.