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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 ()
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 ...
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.)
Derivations of product, quotient, and power rules [ edit ] These are the three main logarithm laws/rules/principles, [ 3 ] from which the other properties listed above can be proven.
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.
A product integral is any product-based counterpart of the usual sum-based integral of calculus. ... Quotient rule (/) ...
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 ′ = ′ ().
Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space; [1] or, more generally, as the quotient of two vectors. [2] A quaternion can be represented as the sum of a scalar and a vector. It can also be represented as the product of its tensor and its versor.