ThmDex – An index of mathematical definitions, results, and conjectures.

Slope function for composition of two differentiable real functions

Formulation 0

Let $f, g : \mathbb{R} \to \mathbb{R}$ each be a D5614: Differentiable real function such that

(i)	\begin{equation} h : \mathbb{R} \to \mathbb{R}, \quad h(x) = f(g(x)) \end{equation}

Then

(1)	$h$ is a D5614: Differentiable real function
(2)	\begin{equation} h'(x) = f'(g(x)) g'(x) \end{equation}

Formulation 1

Let $f, g : \mathbb{R} \to \mathbb{R}$ each be a D5614: Differentiable real function such that

(i)	\begin{equation} h : = f \circ g \end{equation}

Then

(1)	$h$ is a D5614: Differentiable real function
(2)	\begin{equation} \frac{d h(x)}{d x} = \frac{d f(g(x))}{d g(x)} \frac{d g(x)}{d x} \end{equation}