For both partial derivative problems, you can use the chain, product, and power rules, and use z as a variable:
But for 1), you must treat x as a variable and y as a constant, so:
1) first you differentiate both sides with respect to x:
d/dx [ (-5*x + z)^4 - 7*x^6*y^3 + 6*y*z^3 + 2*y^4*z ] = d/dx [ 9 ]
4*(-5x + z)^3 * (-5 + dz/dx) - 42*x^5*y^3 + 6*y*3*z^2*dz/dx + 2*y^4*dz/dx = 0
Then you solve for dz/dx:
-20*(-5x + z)^3 + 4*(-5x + z)^3*dz/dx - 42*x^5*y^3 + 18*y*z^2*dz/dx + 2*y^4*dz/dx = 0
[4*(-5x + z)^3 + 18*y*z^2 + 2*y^4] * dz/dx = 20*(-5x + z)^3 + 42*x^5*y^3
dz/dx = [20*(-5x + z)^3 + 42*x^5*y^3] / [4*(-5x + z)^3 + 18*y*z^2 + 2*y^4]
dz/dx = [10*(-5x + z)^3 + 21*x^5*y^3] / [2*(-5x + z)^3 + 9*y*z^2 + y^4]
and for 2), you must treat y as a variable and x as a constant, so:
2) first you differentiate both sides with respect to y:
d/dy [ (-5*x + z)^4 - 7*x^6*y^3 + 6*y*z^3 + 2*y^4*z ] = d/dy [ 9 ]
Then you solve for dz/dy:
4*(-5*x + z)^3*dz/dy - 7*x^6*3*y^2 + 6*z^3 + 6*3*z^2*dz/dy + 2*4*y^3*z + 2*y^4*dz/dy = 0
4*(-5*x + z)^3*dz/dy - 21*x^6*y^2 + 6*z^3 + 18*z^2*dz/dy + 8*y^3*z + 2*y^4*dz/dy = 0
[ 4*(-5*x + z)^3 + 18*z^2 + 2*y^4 ] * dz/dy = 21*x^6*y^2 - 6*z^3 - 8*y^3*z
dz/dy = [ 21*x^6*y^2 - 6*z^3 - 8*y^3*z ] / [ 4*(-5*x + z)^3 + 18*z^2 + 2*y^4 ]
