We apply the Chain rule to calculate
∂G∂t=∂F∂z∂(At)∂t=∂F∂zA∂G∂γ=∂F∂x∂(γ+s)∂γ+∂F∂y∂(γ−s)∂γ=∂F∂x+∂F∂y∂G∂s=∂F∂x∂(γ+s)∂s+∂F∂y∂(γ−s)∂s=∂F∂x−∂F∂y
For the second derivatives, we apply the Chain rule again to the already calculated derivatives
∂2G∂γ2=∂∂x∂F∂x∂(γ+s)∂γ+∂∂y∂F∂x∂(γ−s)∂γ+∂∂x∂F∂y∂(γ+s)∂γ+∂∂y∂F∂y∂(γ−s)∂γ=∂2F∂x2+∂2F∂y∂x+∂2F∂x∂y+∂F2∂y2∂2G∂s2=∂∂x∂F∂x∂(γ+s)∂s+∂∂y∂F∂x∂(γ−s)∂s−∂∂x∂F∂y∂(γ+s)∂s−∂∂y∂F∂x∂(γ−s)∂s=∂2F∂x2−∂2F∂y∂x−∂2F∂x∂y+∂F2∂y2
Now we set together
∂2G∂γ2+∂G2∂s2=∂2F∂x2+∂2F∂y∂x+∂2F∂x∂y+∂F2∂y2+∂2F∂x2−∂2F∂y∂x−∂2F∂x∂y+∂F2∂y2=2∂2F∂x2+2∂2F∂y2=2∂F∂z
where the last step is already given by the problem.
With ∂2G∂γ2+∂G2∂s2=2∂F∂z and ∂G∂t=A∂F∂z
we conclude that A=2 is the value that ∂2G∂γ2+∂G2∂s2=∂G∂t
