User:IssaRice/Chain rule proofs

Using Newton's approximation

Since $g$ is differentiable at $y_{0}$ , we know $g^{'} (y_{0})$ is a real number, and we can write

$g (y) = g (y_{0}) + g^{'} (y_{0}) (y - y_{0}) + [g (y) - (g (y_{0}) + g^{'} (y_{0}) (y - y_{0}))]$

If we define $E_{g} (Δ y) : = g (y) - (g (y_{0}) + g^{'} (y_{0}) (y - y_{0}))$ we can write

$g (y) = g (y_{0}) + g^{'} (f (x_{0})) (y - y_{0}) + E_{g} (Δ y)$

Newton's approximation says that $| E_{g} (Δ y) | \leq ϵ | y - y_{0} |$ as long as $| y - y_{0} | \leq δ$ .