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Differential Calculus

that branch of mathematics which has for its object the explanation of the method of deriving one determinate function from another by the process of differentiation.

If in any determinate function of one variable we give to the variable a constant increment, and find the corresponding increment of the function, and then divide the increment of the function by the increment of the variable, we shall find a ratio which will in general be dependent upon the increment of the variable. If now we pass to the limit of this ratio, by making the increment of the variable equal to 0, we shall in general obtain a function of the original variable, which is called the differential coefficient of the function. If this be multiplied by the differential of the variable, the result is called the differential of the function. Any function of a single variable will have one, and only one, differential coefficient, and consequently it will have but one differential of the same order. The differential calculus consists of two parts. The first embraces the science of the differential calculus, and explains the methods of finding the differentials and successive differentials of all determinate functions.

The second treats of the application of the differential calculus to the other branches of mathematics, as algebra, analytical geometry, etc. See CALCUJUS.

DIGBY DIFFERENTIAL THERMOMETER, an instrument for determining very minute differences of temperature. Leslie's differential thermometer consists of two glass bulbs containing air connected by a bent tube containing some sulphuric acid, the movement of which (as the air expands and contracts) serves to indicate any slight difference of temperature between the two bulbs.