The output of the nation's factories, mines and utilities, which was growing swiftly late last year, fell sharply in January as natural gas shortages and cold weather stymied production, the Federal Reserve Board reported yesterday.

The 1 per cent decline in the industrial production index is the biggest one-month drop since output fell more than 2 per cent in February, 1975, during the depths of the recession.

Further declines in industrial output are expected in February because of the cold weather, government officials said.

Changes in industrial production translate quickly into jobs as factories lay off or recall workers. The Commerce Department said that the number unemployed because of shortages of natural gas reached a peak of about 1.25 million people in early February.

The most recent survey indicates that about 500,000 workers remain unemployed because of the natural gas shortage.

The January industrial production index was the first major government economic statistic to reflect the impact of the frigid temperatures and the natural gas shortages the weather helped induce in the Eastern two-thirds of the nation.

Industrial output rose 0.8 per cent in December and 1 per cent in November. Federal Reserve Board analysts said were it not for the cold weather the index would have risen in January as well, although they could not say by how much.

Commerce Department chief economist John W. Kendrick said that manufacturing is one of the three major areas to be affected strongly by the weather and the gas shortages. The others are construction and agriculture.

Kendrick said his agency expects that after another decline in February, factories should increase production and hiring in March. He acknowledges that if the weather is much colden than normal during late February and March his prediction could be wrong.

Earlier the Commerce Department had estimated that the weather would knock about 2 percentage points off the overall economic growth rate (the increase in the so-called real gross national product) during the first three months of the year.

Most economists had predicted that the amount of goods and services produced by American citizens would rise 6 per cent during the first quarter. If the Commerce Department estimates are correct, the figure would drop to 4 per cent.

Carter administration officials, such as Office of Management and Budget Director Bert Lance, have said that they think the impact of th ecold weather will be transitory, with much of the production lost now made up in the coming months.

The Federal Reserve said the biggest production decline came among consumer durable goods. Automobile assemblies declined 15 per cent during January to an annual rate of 8.5 million cars.

Gas and electric utilities increased their output so much because of the cold weather that they offset declines in production of other consumer goods. There are only a 0.1 per cent decline in production of nndurable consumer goods.

Output of business equipment fell 0.5 per cent, while production of materials such as paper, textiles and chemicals was off 1.2 per cent.

Industrial production accounts for about a third of the total gross national product. The Federal Reserve Board said that the worst of the winter weather and gas shortages occurred in the last half of January.

The agency emphasized that since the information about the effects of the weather is incomplete the January report "is considered more tentative than usual."

Earlier government statistics on unemployment and wholesale prices were collected befoer most of the cold weather and fuel shortages struck and reflected the improving economy in late 1976 and the early weeks of 1977.

The unemployment rate fell from 7.8 per cent to 7.3 per cent in January and wholesale prices rose 0.5 per cent, after rising 0.6 per cent in December.