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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 ...
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
Often the power rule, stating that () =, is proved by methods that are valid only when n is a nonnegative integer. This can be extended to negative integers n by letting n = − m {\displaystyle n=-m} , where m is a positive integer.
The rule can be thought of as an integral version of the ... The antiderivative of − 1 / x 2 can be found with the power rule and is ... The proof uses the ...
5.1 Proof for Taylor's theorem in one real variable. 5.2 Alternate proof for Taylor's theorem in one real variable. ... By the Power Rule, repeated derivatives of ...
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The proof uses the comparison test, comparing the term () with the integral of ... (cf. Riemann zeta function) converges for every ε > 0, because by the power rule
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