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??
Of course, you would want to cancel the "xy" terms in numerator and denominator of that last fraction to get
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.