A first immediate consequence of the definition is that

In this situation, the chain rule represents the fact that the derivative of is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

As a consequence of this demonstration, Descartes considers science and mathematics to be justified to the extent that their proposals are established on a similarly immediate clarity, distinctiveness, and self-evidence that presents itself to the mind.

For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.

His grandfather, Thomas Francis, founder of the Carignano line of the House of Savoy, was the son of Catherine Michelle – a daughter of Philip II of Spain – and the great-grandson of the Emperor Charles V. But of more immediate consequence to Leopold I was the fact that Eugene was the second cousin of Victor Amadeus, the Duke of Savoy, a connection that the Emperor hoped might prove useful in any future confrontation with France.

An immediate consequence of this is that R < sup > m </ sup > is not homeomorphic to R < sup > n </ sup > if m ≠ n — an intuitively " obvious " result which is nonetheless difficult to prove.

One consequence not foreseen by the builders of the system was that with the almost immediate reporting of newsworthy events, tens of thousands of people worldwide — along with criminals — would flock to the scene of anything interesting, hoping to experience or exploit the instant, thus disorder and confusion be created.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

Its immediate consequence was a large Protestant exodus from France.

An immediate and important consequence of this distinguishing feature is that in the relational model the Cartesian product becomes commutative.

The immediate consequence of William's death was a war between his sons Robert and William over control of England and Normandy.

One immediate consequence was the replacement of the curving bat with the straight one.

The immediate consequence was that the rule of the Asturians passed to his brother-in-law, ruler of the neighboring independent domain, through a marriage alliance to Fafila's sister.

As an immediate consequence of the London conference 1948 the representatives of the three western occupations powers on the 1st of July convoked the Ministerpräsidenten ( prime ministers ) of the West German Länder in Frankfurt / Main and committed to them the socalled Frankfurt Documents ( Frankfurter Dokumente ).

Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate consequence.

An immediate consequence of the definition is that every principal ideal domain ( PID ) is a Dedekind domain.

The immediate consequence of Clive's victory at Plassey was an increase in the revenue demand on Bengal by at least 20 %, much of which was appropriated by Zamindars and corrupt Company Officials, which led to considerable hardship for the rural population, particularly during the famine of 1770.

As noted by Jerzy Kowalski-Glikman, an immediate consequence of these postulates is that the symmetry group of DSR theories must be ten dimensional, corresponding to boosts, rotations and translations in 4 dimensions.

The immediate consequence is that it is demonstrated to have high reliability.

Its immediate consequence in Sweden proper was the deposition of Gustav IV on March 13, 1809, and the subsequent exclusion of his whole family from the succession.

The immediate consequence of Deng's price reform was a sudden and massive inflation, unprecedented in the experience of the younger generation and particularly frightening to older folks who could still remember the rampant inflation in the last years of the Nationalist regime.

The immediate consequence of the Constitution was the opening of the first Parliamentary government in Asia.

This has the immediate consequence that the set of zeros of f, near ( 0, ..., 0 ), can be found by fixing any small value of z and then solving W ( z ).

In 1988, the Holy See declared that Williamson had incurred automatic excommunication latae sententiae, i. e. as a direct and immediate consequence of his unauthorized consecration as a bishop by Archbishop Marcel Lefebvre.

But when the situation was so complicated that even Nogaret, one of the principal actors in the drama, could misinterpret the pope's motives, it is possible that Othon and his companions, equally baffled, attributed their difficulties to a more immediate cause.

when his Holiness Pope John 23, first called for an Ecumenical Council, and at the same time voiced his yearning for Christian unity, the enthusiasm among Catholic and Protestant ecumenicists was immediate.

These never ceased to suggest that if, in the eyes of Marx and Lenin `` full communism '' was still a very distant ideal, the establishment of a Communist society had now, under Khrushchev, become an `` immediate and tangible reality ''.

There was no reference to the incident on the stairs, his powers being absorbed by this more immediate business.

Louise climbed onto a stool and clutched the hand with which I was trying to hold the phone, claiming my immediate attention on grounds of extreme emergency.

Uncle John Vinnicum Morse was the immediate popular suspect.

She had quarreled with Lucien, she had resisted his demands for money -- and if she died, by the provisions of her marriage contract, Lucien would inherit legally not only the immediate sum of gold under the floorboards in the office, but later, when the war was over, her father's entire estate.

But she didn't sound real alarmed you know, like there was any immediate danger.

When he was unable to bring about immediate expansion, he sought to convince another National League club to move here.

So, too, was the insistence on the relativity of the external world, and the ideas that language and things perceived by consciousness were poor substitutes indeed for immediate perception by pure, indwelling spirit: the opposition of pure consciousness to ratiocinating consciousness.

the purpose was less immediate.

The immediate influence of Aristotle's work was felt as the Lyceum grew into the Peripatetic school.

Johnston's initial call upon the governors for more men did not result in many immediate recruits but Johnston had another, even bigger, problem since his force was seriously short of arms and ammunition even for the troops he had.

Algardi was also known for his portraiture which shows an obsessive attention to details of psychologically revealing physiognomy in a sober but immediate naturalism, and minute attention to costume and draperies, such as in the busts of Laudivio Zacchia, Camillo Pamphilj, and of Muzio Frangipane and his two sons Lello and Roberto.

He had been suffering from tuberculosis, but the immediate cause of his death was a recurrence of dysentery.

The drive was smoother than that of the previous day, as the craters were more shallow and boulders were less abundant north of the immediate landing site.

The senator Falier was induced to receive him under his immediate protection.

The results of this work was the report A Public Trust at Risk: The Heritage Health Index Report on the State of America's Collections, which was published in December 2005 and concluded that immediate action is needed to prevent the loss of 190 million artifacts that are in need of conservation treatment.

The feature was launched on Monday, August 13, 1934 in eight North American newspapers — including the New York Mirror — and was an immediate success.

The majority of bombing was still done by one-engined biplanes with one or two crew-members flying short distances to attack the enemy lines and immediate hinterland.

Soviet-style centralised planning in five-year blocks had more immediate benefits there than in the other European states where it was first applied in the early 1950s.

This led to some immediate international trade liberalization, but there was no shock to the economy.

Upon boot, the user was directed to a BASIC interpreter in immediate mode, not a traditional command line interface.