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The power rule for differentiation was derived by Isaac Newton and Gottfried Wilhelm Leibniz, each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse operation. This mirrors the conventional way the related theorems are presented in modern basic ...
Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each connected component of (those constants need not be equal). This observation implies that if a function g : U → C {\displaystyle g:U\to \mathbb {C} } has an antiderivative, then that antiderivative is unique up to addition of a function which ...
The antiderivative of − 1 / x 2 can be found with the power rule and is 1 / x . Alternatively, one may choose u and v such that the product u ′ (∫ v dx ) simplifies due to cancellation.
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g, ′ = () ′ = (′ + ′ ), wherever both sides are well defined. Special cases
The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. dot product In mathematics , the dot product or scalar product [ note 1 ] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors ) and returns a single number.
They imply the power rule = In addition, various forms of the chain rule hold, in increasing level of generality: [12] If y = f ( u ) is a differentiable function of the variable u and u = g ( x ) is a differentiable function of x , then d y = f ′ ( u ) d u = f ′ ( g ( x ) ) g ′ ( x ) d x . {\displaystyle dy=f'(u)\,du=f'(g(x))g'(x)\,dx.}
closed forms, i.e., dω = 0, have zero integral over boundaries, i.e. over manifolds that can be written as ∂Σ c M c, and; exact forms, i.e., ω = dσ, have zero integral over cycles, i.e. if the boundaries sum up to the empty set: ∂Σ c M c = ∅. De Rham's theorem shows that this homomorphism is in fact an isomorphism. So the converse to ...