Partial Derivatives of f(r,a,b) and Solving for r,a,b in Terms of x,y,z

[MathematicalPhysicist]

im given the function f(r,a,b) and z=rcos(a) y=rsin(a)sin(b) x=rsin(a)cos(b)
now i need to find the partial derivative of f'_y, without solving r,a,b in terms of x,y,z, what that i got is:
f'_y=f'_a*a'_y+f'_r*r'_y+f'_b*b'_y
the answer should include the derivatives of f wrt r,a,b, which i think i did ok, (i only needed to calculate a'_y,b'_y,r'_y).
but i have another question and it's to solve r,a,b in terms of x,y,z which i did, and then to calculate the above derivative directly by the chain rule.
i think i need to use here a jacobian, so i defined the implicit functions: F=z-rcos(a) G=y-rsin(a)sin(b) H=x-rsin(a)cos(b), my proble is that in order to calclualte it: i need to know what's f, cause:
f'_y=-J((F,G,H)/(x,y,z))/J((F,G,H)/(x,f,z))
the question is how to do it?

thanks in advance.

 

[Quadruple Bypass]

cant u use chain rule?

 

[HallsofIvy]

You would use the Jacobian in changing variables in integration.

To differentiate in different variables, use the chain rule.

 

[MathematicalPhysicist]

so how would i calclulate the derivatives of:
f'_a f'_r and f'_b, i mean after i solved for a,r,b in terms of x,y,z:
f'_y=f'_r*r'_y+f'_a*a'_y+f'_b*b'_y
and
f'_r=f'_x*x'_r+f'_y*y'_r+f'_z*z'_r
and then to substitue f'_r in the first equation and this way also to do with f'_b and f'_a, is this correct?

 

[HallsofIvy]

Yes, that is correct.

And it is easy to see that $r= \sqrt{x^2+ y^2+ z^2}$, $b= arctan(\frac{y}{x})$, and $a= arctan(\frac{\sqrt{x^2+y^2}}{z}$.