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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 ...
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The most general power rule is the functional power rule: for any functions and , ′ = () ′ = (′ + ′ ), wherever both sides are well defined. Special cases: If f ( x ) = x a {\textstyle f(x)=x^{a}} , then f ′ ( x ) = a x a − 1 {\textstyle f'(x)=ax^{a-1}} when a {\textstyle a} is any nonzero real number and x {\textstyle x} is ...
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The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the ...
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Not endlessly many. And if you are going to give examples, then you need examples of integration as well as differentiation. Deleted more strange stuff; These results can be verified with an understanding of Newton's difference quotient and the binomial theorem. One can also derive the General Power Rule via the Chain Rule.
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