Differential Calculus | Research & Encyclopedia Articles

Differential Calculus

Elementary calculus is usually divided into two branches, differential calculus and integral calculus. Differential calculus is the branch of calculus that is based on the determination of the limit of a certain ratio whereas integral calculus is based on the determination of the limit of a certain sum. Differential calculus is the portion of calculus that deals with derivatives. The derivative of a function is representative of an infinitesimal change in the function with respect to its parameters. The derivative of a function f with respect to x is denoted either as f'(x) or df/dx. Leibniz developed this notation in 1684. All applications of differential calculus are concerned with interpretations of the derivative as the slope of the line tangent to the curve at a specific point or as the rate of change of the dependent variable with respect to the independent variable. Employing differential calculus provides a method for determining the slope of a line tangent to a curve, rates of change, points moving on a straight line or other curve, and absolute maxima and minima. It is used by the physical and biological sciences as well as statistical analysis used in business and social studies.

In order to describe differential calculus as the determination of the limit of a certain ratio we must first understand the ratio. Let f(x) = y where y is the dependent variable that is a function of the independent variable x. If x₀ is a value of x defined in the domain identified for x, then y₀ = f(x₀) is the corresponding value of y. Let h and k be real numbers, and y₀ + k = f(x₀ + h). So k = f(x₀ + h) - f(x₀) and k/h = [f(x₀ + h) - f(x₀)]/h. This ratio is the difference quotient and equals the tangent of the curve drawn between the points (x₀, y₀) and (x₀ + h, y₀ + k). The difference quotient can be thought of as the average rate of change of y = f(x) with respect to x within the defined interval. If the limit of this ratio k/h exists as h approaches 0 then this limit is called the derivative of y with respect to x, evaluated at x = x₀. The limit is written as lim_{h 0} k/h. This derivative can be interpreted as the slope of the curve y = f(x) at x = x₀.

It can also be interpreted as the instantaneous rate of change of y with respect to x at x₀. Rules and methods developed by this limit process enabled mathematicians to formulate various equations that provide for rapid calculation of the derivatives of various functions. The derivative of a function f(x) of x is usually denoted as f'(x). When the derivative of f(x) is found for all values of x a new function is obtained which is itself a function of x. The derivative of f'(x) can be found and is called the second order derivative of y with respect to x and is usually denoted as f''(x), (d²y)/(dx²) or (d²f(x))/(dx²). Higher order derivatives are expressed in the same manner.

Isaac Newton and Gottfried Wilhelm Leibniz are attributed with inventing calculus in the 17th century. Although these two are credited with inventing calculus isolated results concerning its fundamental problems had been known for thousands of years. The Egyptians used calculus to determine the volume of a pyramid and the area of a circle. Ancient Greek scientists employed calculus to study the stars and planetary motion long before calculus was formally recognized. By the early 17th century mathematicians had developed methods for determining the areas and volumes of a wide variety of shapes and forms. In about 1720 Isaac Barrow published a book stating geometrically the inverse relationship between problems of finding tangents and areas. This relationship is now known as the fundamental theorem of calculus. In 1665-1666 Newton's discovery of combining infinite sums or infinite series, the binomial theorem for fractional exponents, and the algebraic expression of the inverse relationship between tangents and areas led his to develop methods used in calculus. He was reluctant to publish these results and so Leibniz published his discovery of differential calculus in 1684. Although Leibniz developed his methods of calculus independently of Newton and so much later than Newton he is still recognized as the codiscoverer of calculus since he published his results. Leibniz also replaced Newton's symbols in differential calculus with those that are used today. Calculus is used in the physical sciences to study the speed of a falling object, the rates of change in a chemical reaction, and the rate of decay of a radioactive material. The biological sciences employ calculus to study problems such as the rate of growth of a colony of bacteria as a function of time. The social sciences use calculus to study problems concerned with statistics and probability, such as population.