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In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and ()
The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable but only says what its derivative is if it is differentiable.)
2.4 Quotient rule for division by a scalar. 2.5 Chain rule. 2.6 Dot product rule. ... We have the following generalizations of the product rule in single-variable ...
1.3 The product rule. 1.4 The chain rule. 1.5 The inverse function rule. 2 Power laws, ... The reciprocal rule can be derived either from the quotient rule, or from ...
Then can be written (by the product rule) as + where now denotes the multiplication operator by the logarithmic derivative ′ In practice we are given an operator such as D + F = L {\displaystyle D+F=L} and wish to solve equations L ( h ) = f {\displaystyle L(h)=f} for the function h , given f .
Also, one can readily deduce the quotient rule from the reciprocal rule and the product rule. The reciprocal rule states that if f is differentiable at a point x and f(x) ≠ 0 then g(x) = 1/f(x) is also differentiable at x and ′ = ′ ().
Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V .
Discrete differential calculus is the study of the definition, properties, and applications of the difference quotient of a function. The process of finding the difference quotient is called differentiation.