Production at the nation's factories, mines and utilities shot up 0.7 percent in April, the biggest gain in six months, the government reported today. The Federal Reserve said the April advance followed increases of 0.4 percent in January and 0.2 percent in March and no change at all in February. It was the largest advance since a 1.1 percent surge last October. The increase was in line with expectations as economists had predicted a strong showing based on a pickup in production at auto factories. Autos were assembled at an annual rate of 7 million units last month, up from a rate of 6.6 million units in March. With the latest improvement, overall industrial production is now 6.4 percent higher than it was a year ago, a further sign of the resurgence of the U.S. manufacturing sector, which has enjoyed booming export sales because of the decline in the value of the dollar. The Reagan administration is counting on further reductions in the U.S. trade deficit to contribute almost half of the expected growth in the overall economy this year. The April advance in industrial output followed a period of generally sluggish activity in the first part of the year as production gains slowed while manufacturers, particularly in the auto industry, tried to work down an overhang of unsold goods. Still, the production slowdown was much less severe than had been feared immediately after the October stock market collapse. At that time, some economists believed that a drop in consumer spending would force such steep cuts in production that the country would be pushed into a recession. Instead, the overall economy expanded at a moderate 2.3 percent annual rate in the first three months of the year, as measured by the gross national product. The April report showed that output in the manufacturing sector climbed 0.7 percent in April, reflecting an increase of 0.9 percent in the production of durable goods, items expected to last three years or more, and a 0.4 percent rise in non-durable production. Output in the mining industry jumped 1.3 percent in April, following a 0.7 percent March increase. The strong gains in both of those months followed a string of declines. This sector, which includes oil and gas drilling, has been depressed because of falling oil prices. Output at the nation's utilities fell 0.8 percent in April, following an even larger 1.1 percent drop in March. The various changes left the industrial production index at 135.6 in April, up from 134.7 in March.