Proposition 5

Let $f$ be a real function defined on an interval $I \subset \mathbf{R}$, and $g$ a vector function defined on an interval of $\mathbf{R}$ containing $f(I)$. If $f$ is differentiable at the point $x_0$ and $g$ is differentiable at the point $f(x_0)$ then the composite function $g \circ f$ has a derivative equal to $g'(f(x_0)) f'(x_0)$ at $x_0$.

Let us put $h = g \circ f$; for $x \neq x_0$ we can write

where we set $u(x) = \frac{g(f(x)) - g(f(x_0))}{f(x) - f(x_0)}$ if $f(x) \neq f(x_0)$, and $u(x) = g'(f(x_0))$ otherwise. Now $f(x)$ has limit $f(x_0)$ when $x$ tends to $x_0$, so $u(x)$ has limit $g'(f(x_0))$, from which the proposition follows in view of the continuity of the function $yx$ on $E \times \mathbf{R}$.