Differential calculus is a subfield of differential calculus that focuses on derivatives, which are used to describe rates of change that are not constant. The term "differential" comes from the process known as differentiation, which is the process of finding the derivative of a curve. Differential calculus is an important topic covered in differential calculus. According to Interactive Mathematics, "We use the derivative to determine the maximum and minimum values ​​of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.)." Not only are derivatives used to determine how to maximize or minimize functions, but they are also used to determine how two related variables change over time relative to each other. Eight different differential rules have been established to help find the derivative of a function. Such rules include the chain rule, differentiating the sum and difference of equations, the constant rule, the product rule, the quotient rule, and more. In addition to these differential rules, optimization is an application of differential calculus used today to effectively help with efficiency. Additionally, partial differentiation and implicit differentiation are subgroups of differential calculus that allow you to bring derivatives into more challenging and difficult formulas. The mean value theorem is applied in differential calculus. This rule essentially states that there is at least one tangent line that produces the same slope as the slope formed by the endpoints found on a closed interval. Differential calculus began to develop because of Sir Isaac Newton's greatest problem: navigation at sea. Shipwrecks were frequent, all due to the fact that the captain was not aware of how the Earth, planets and stars move... middle of paper... mathematics, would not be able to exist to the extent which it is today.Works CitedWilliam, Walter and Rouse Ball. Differentiation rules: chain rule, inverse functions and derivation, sum rule in derivation, constant factor rule in derivation. New York City, NY: General Books LLC, 1888. Print.Kouba, Duane. "Problems of implicit differentiation". Collection of lessons. (1998): Print.Rusin, Dave. "Partial Differential Equations". Mathematical atlas. 35.1 (2000): Print.Foster, Niki. "Who is Gottfried Leibniz." Short and simple guide (2011): n. page Network. April 14, 2011. Bourne, M. “Applications of Differentiation.” Interactive mathematics. Np, 25 02 2011. Web. 14 April 2011. .