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This leads directly to the notion that the differential of a function at a point is a linear functional of an increment . This approach allows the differential (as a linear map) to be developed for a variety of more sophisticated spaces, ultimately giving rise to such notions as the Fréchet or Gateaux derivative.
Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from R m {\displaystyle \mathbb {R} ^{m}} to R n {\displaystyle \mathbb {R} ^{n}} .
The exterior derivative is a notion of differentiation of differential forms which generalizes the differential of a function (which is a differential 1-form). Pullback is, in particular, a geometric name for the chain rule for composing a map between manifolds with a differential form on the target manifold.
A differentiable function. In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.
For any functions and and any real numbers and , the derivative of the function () = + with respect to is ′ = ′ + ′ (). In Leibniz's notation , this formula is written as: d ( a f + b g ) d x = a d f d x + b d g d x . {\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
Differential geometry, exterior differential, or exterior derivative, is a generalization to differential forms of the notion of differential of a function on a differentiable manifold; Differential (coboundary), in homological algebra and algebraic topology, one of the maps of a cochain complex; Differential cryptanalysis, a pair consisting of ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).