Use the sufficient conditions for differentiability to determine where the function $f(z) = e^{z^2}$ is differentiable

Question

I know that the exponential function satisfies the condition of Cauchy-Riemann and is differentiable but how to tell where the function is differentiable specifically? Sufficient Conditions for Differentiability in this case are: If the real functions u(x,y) and v(x, y) are continuous and have continuous first-order partial derivatives in some neighborhood of a point z, and if u and v satisfy the Cauchy-Riemann equations at z, then the complex function f(z) = u(x, y) + iv(x, y) is differentiable at z

Answer 1

That function can be naturally expressed as the composition of two differentiable maps (the exponential function and $z\mapsto z^2$). Therefore, it is differentiable. You don't need the Cauchy-Riemann equations here.