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I'm not sure that I would call $\partial z/\partial x$ the gradient in this context but, assuming your condition is on $\partial z/\partial x$, you could do something like this: eq = x^2 + y^2 + z^2 == 1; deriv = Derivative[1, 0][z][x, y] /. First[ Solve[D[eq /. z -> z[x, y], x], Derivative[1, 0][z][x, y]]] /. z[x, y] -> z (* Out: -x/z *) Note ...



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