Im working on the derivative f'y(x,y) of the function f(x,y)=arctan(x^2)+ y^3
Because it's a derivative with respect to the variable y, this means that the variable x is held constant, does this mean that the entire arctan(x^2) should be regarded as just a number so the partial derivative equals to 3y^2 in the end??
You could see that more easily by writing f(x, y, z()= 2(z+ 3)^4+ ln(xyz)= 2(z+ 3)^4+ ln(x)+ ln(y)+ ln(z).
Oh, and don't write "f'z" for both the function and its partial derivative!
In fact, it is not a good idea to use the ' with partial derivatives. You problem is to find the derivative, with respect to z,
of f(x,y,z)= 2(z+ 3)^4+ ln(xyz) and the correct answer, in simplest form, is f_z= 8(z+ 3)^3+ 1/z.