Leibniz would express the Chain Rule as:
The Chain Rule is a formula for the derivative of the composition of two functions. Suppose the realvalued function g(x) is defined on some open subset, of the real numbers, containing the number x; and h[g(x)] is defined on some open subset of the reals containing g(x). If g is differentiable at x and h is differentiable at g(x), then the composition h o g is differentiable at x and the derivative can be computed as

Example I Consider f(x) = (x^{2} + 1)^{3}. f(x) is comparable to h[g(x)] where g(x) is (x^{2} + 1) and h(x) is x^{3}; thus, f '(x) = 3(x^{2} + 1)^{2}(2x) = 6x(x^{2} + 1)^{2}.
Example II In order to differentiate the trigonometric function:
The General Power Rule The General Power Rule (GPR) is derivable, via the Chain Rule.
The Fundamental Chain Rule The chain rule is a fundamental property of all definitions of derivative and is therefore valid in much more general contexts. For instance, if E, F and G are Banach spaces (which includes Euclidean space) and f : E > F and g : F > G are functions, and if x is an element of E such that f is differentiable at x and g is differentiable at f(x), then the derivative of the composition g o f at the point x is given by
A particularly nice formulation of the chain rule can be achieved in the most general setting: let M, N and P be C^{k} manifolds (or even Banachmanifolds) and let f : M > N and g : N > P be differentiable maps. The derivative of f, denoted by df, is then a map from the tangent bundle of M to the tangent bundle of N, and we may write
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