Or should I write this as differentiate respect to x only in my example.
and
If you are using implicit differentiation, that is different from partial differentiation, and you need a bit more:
And you already have the other formula:
Now, you can set these equal to each other, but you won't get .
Edit: But .
Differentiating we get
Solving for , we get
Now, you can plug that in:
Now, setting those equal, you can solve for .
Edit 2: But, setting those equal just yields the identity .
Thanks, I am not trying to use implicit differentiation. I just try to right it out in the way it make sense.
I know in partial differentiation, I only use one of either x or y as a variable in rectangle coordinates, and use either r or in polar coordinates. Then treat the other variable as constant. That's the reason:
where as y is treated as constant.
How should I write it if I want to treat ?
Again, thanks for your help.
I rechecked your posts. I don't understand what example you mean. There was no example given. This whole thread seems to be devoted to trying to find simple formulas for the derivatives of the change of coordinates formulas when you hold certain variables constant.
Well, if you fix , then . If is 60 degrees (as you suggested), then , just as I calculated in post #4.
If you don't fix any of the variables, then the partial derivatives may not be zero.
I don't know what you are looking for then. The "if"s are necessary. Whatever differential formula you are pointing to either looks at a special case (when some variable is fixed), or it is more complex than what you want to hear. But, I can't help you force the mathematics to do what you want. The mathematics works just as I showed you. Either the partial derivatives are fixed or they are not. If they are fixed, you get the formula you wrote in the original post. If they are not fixed, then you don't get the formula you wrote in your original post. So, whatever answer you are looking for may not exist.