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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 ...
Simplest rules Derivative of a constant; Sum rule in differentiation; Constant factor rule in differentiation; Linearity of differentiation; Power rule; Chain rule; Local linearization; Product rule; Quotient rule; Inverse functions and differentiation; Implicit differentiation; Stationary point. Maxima and minima; First derivative test; Second ...
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.
The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule. For example, from the identity A ⋅( B × C ) = ( A × B )⋅ C we may derive A ⋅(∇× C ) = ( A ×∇)⋅ C but not ∇⋅( B × C ) = (∇× B ...
Power rule. differential of x n; Product and Quotient Rules; Derivation of Product and Quotient rules for differentiating. Prime number. Infinitude of the prime numbers; Primitive recursive function; Principle of bivalence. no propositions are neither true nor false in intuitionistic logic; Recursion; Relational algebra (to do) Solvable group ...
This is known as the power rule. For example, () = =. However, many other functions cannot be differentiated as easily as polynomial functions, meaning that sometimes further techniques are needed to find the derivative of a function.
In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.
The derivative of the function given by () = + + is ′ = + () () + = + (). Here the second term was computed using the chain rule and the third term using the product rule. The known derivatives of the elementary functions , , (), (), and =, as well as the constant , were also used.