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Reciprocal rule. In calculus, the reciprocal rule gives the derivative of the reciprocal of a function f in terms of the derivative of f. The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents. Also, one can readily deduce the quotient rule from the ...
This follows from the difference-quotient definition of the derivative. The last equality follows from the continuity of the derivatives at c. The limit in the conclusion is not indeterminate because ′ (). The proof of a more general version of L'Hôpital's rule is given below. General proof
The derivative of () = for any (nonvanishing) function f is: ′ = ′ (()) wherever f is non-zero. In Leibniz's notation, this is written (/) =.The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule.
Calculus. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. [1] [2] [3] Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is. It is provable in many ways by using other derivative rules .
In mathematics, the Fréchet derivative is a derivative defined on normed spaces.Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations.
For a given arbitrary stencil points of length with the order of derivatives , the finite difference coefficients can be obtained by solving the linear equations [5] where is the Kronecker delta, equal to one if , and zero otherwise. Example, for , order of differentiation :
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,
The Lie derivative is the rate of change of a vector or tensor field along the flow of another vector field. On vector fields, it is an example of a Lie bracket (vector fields form the Lie algebra of the diffeomorphism group of the manifold). It is a grade 0 derivation on the algebra.