Defina
g = v - w^2. Então
Usando a regra do produto e a cadeia:
\frac{\partial R}{\partial x} = \frac{\partial u}{\partial x}\,e^{g} + u\,e^{g}\,\frac{\partial g}{\partial x}, \qquad \frac{\partial R}{\partial y} = \frac{\partial u}{\partial y}\,e^{g} + u\,e^{g}\,\frac{\partial g}{\partial y}.
Agora,
\frac{\partial u}{\partial x}=1,\quad \frac{\partial u}{\partial y}=2,
\frac{\partial v}{\partial x}=1,\quad \frac{\partial v}{\partial y}=-1,
\frac{\partial w}{\partial x}=1,\quad \frac{\partial w}{\partial y}=1.
Logo,
\frac{\partial g}{\partial x} = \frac{\partial v}{\partial x} - 2w\frac{\partial w}{\partial x} = 1 - 2w,
\frac{\partial g}{\partial y} = \frac{\partial v}{\partial y} - 2w\frac{\partial w}{\partial y} = -1 - 2w.
Substituindo em cada derivada:
\frac{\partial R}{\partial x} = e^{g} + u\,e^{g}\,(1 - 2w) = e^{g}\bigl[1 + u(1 - 2w)\bigr],
\frac{\partial R}{\partial y} = 2\,e^{g} + u\,e^{g}\,(-1 - 2w) = e^{g}\bigl[2 + u(-1 - 2w)\bigr].