Differentiation rules
The chain rule

The derivative of the function h (x) = f (g (x)) {\displaystyle h(x)=f(g(x))} is h ′ (x) = f ′ (g (x)) ⋅ g ′ (x). {\displaystyle h'(x)=f'(g(x))\cdot g'(x).}

In Leibniz's notation, this is written as: d d x h (x) = d d z f (z) | z = g (x) ⋅ d d x g (x), {\displaystyle {\frac {d}{dx}}h(x)=\left.{\frac {d}{dz}}f(z)\right|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),} often abridged to d h (x) d x = d f (g (x)) d g (x) ⋅ d g (x) d x. {\displaystyle {\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.}

Focusing on the notion of maps, and the differential being a map D {\displaystyle {\text{D}}}, this is written in a more concise way as: x = g (x) ⋅ x. {\displaystyle _{x}=_{g(x)}\cdot _{x}\,.}
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