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The chain rule holds whenever both the inner function $h$ is differentiable at $x$ and the outer function $g$ is differentiable at $h(x)$. If $h'(x)=0$, the formula $g'(h(x))h'(x)$ correctly yields 0.

Conditions: $h$ is differentiable at $x$; $g$ is differentiable at $h(x)$; No requirement for $h'(x) \neq 0$

The chain rule holds whenever both the inner function $h$ is differentiable at $x$ and the outer function $g$ is differentiable at $h(x)$. If $h'(x)=0$, the formula $g'(h(x))h'(x)$ correctly yields 0.

Conditions: $h$ is differentiable at $x$; $g$ is differentiable at $h(x)$; No requirement for $h'(x) \neq 0$