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.

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.

Mathers sent an immediate reply, declining to provide proof, refusing to acknowledge the London temple, and dismissing Farr as his representative on March 23.

In the immediate postwar years housing was a major issue in London, due to the large amount of housing which had been destroyed in the war.

Shortly afterwards, false reports spread in mid-March 1939 by the Romanian minister in London, Virgil Tilea, that his country was on the verge of an immediate German attack, led to a dramatic U-turn in British policy of resisting commitments in Eastern Europe.

The London Sustainable Development Commission has calculated that for housing to meet the 60 % target, all new developments would have to be constructed to be carbon-neutral with immediate effect ( using zero energy building techniques ), in addition to cutting energy used in existing housing by 40 %.

When it was performed in London in 1862, it was an immediate sensation.

Revised and expanded, it was performed at the Crystal Palace in 1862, a year after his return to London, and was an immediate sensation.

Pirates was an immediate hit in New York, and later London, becoming one of the most popular Gilbert and Sullivan operas.

When Adam Bamme, the mayor of London, died in June 1397, Whittington was imposed on the city by the King as Lord Mayor of London two days later to fill the vacancy with immediate effect.

An early NME article on the band wrote that The Cure " are like a breath of fresh suburban air on the capital's smog-ridden pub-and-club circuit " and noted " With a John Peel session and more extensive London gigging on their immediate agenda, it remains to be seen whether or not The Cure can retain their refreshing joie de vivre.

In its own time, the London and national papers shared the horror felt in the Manchester region, but Peterloo's immediate effect was to cause the government to crack down on reform, with the passing of what became known as the Six Acts.

Limehouse station, located to the immediate west of the Limehouse Basin, serves the c2c mainline trains out to Essex, and Docklands Light Railway, which is two stops from Bank & Tower Gateway in the City of London.

Six trio sonatas were the other immediate fruits of his time in London.

It was attacked in the first major air-raid over the London area and factories in its immediate vicinity were almost destroyed with the loss of six airmen and over 60 civilians.

In 1772, Richard Sheridan, at the age of 21, eloped with and subsequently married Elizabeth Ann Linley and set up house in London on a lavish scale with little money and no immediate prospects of any — other than his wife's dowry.

The original production opened in London at Covent Garden Theatre on 15 March 1773 and was an immediate success.

No immediate work was carried out on the deep-level line, and the subsequent take over of the MDR by the Underground Electric Railways Company of London ( UERL ) and the resignalling and electrification of the MDR's routes between 1903 and 1905 meant that congestion was relieved without needing to construct the deep-level line.

She had immediate success in both London and Paris.

At the Restoration, having shared Charles's banishment, Sir George formed one of the immediate train of the restored monarch on his triumphant entry into London.

Transport for London made the ending of the tidal flow with immediate effect without advance notice.

Regularly scheduled electronically scanned television began from Alexandra Palace in London on 2 November 1936, to just a few hundred viewers in the immediate area.

In 1680 William Dockwra established the London Penny Post, a mail delivery system that delivered letters and parcels weighing up to one pound within the city of london and some of its immediate suburbs for the sum of one penny .< ref name =" William Dockwra and the Penny Post Service "></ span ></ font ></ ref >

It is one of the very few stations served by London Underground which has no ticket gates and due to the restrictive layout here there are no plans for these to be installed in the immediate future.

No immediate work was carried out on the deep-level line, and the subsequent take over of the MDR by the Underground Electric Railways Company of London ( UERL ) and the resignalling and electrification of the MDR's routes between 1903 and 1905 meant that congestion was relieved without needing to construct the deep-level line.