In mathematics, differential calculus is a subfield of calculus concerned with the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential. The derivative of a function at a chosen input value describes the behavior of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In general, the derivative of a function at a point determines the best linear approximation to the function at that point.
The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.
Differentiation has applications to all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. By applying game theory, differentiation can provide best strategies for competing corporations.
Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.
HISTORY OF DIFFERENTIAL CALCULUS
The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. 300 BCE), Archimedes (c. 287–212 BCE) and Apollonius of Perga (c. 262–190 BCE).Archimedes also introduced the use of infinitesimals, although these were primarily used to study areas and volumes rather than derivatives and tangents; see Archimedes' use of infinitesimals.
The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as early as 500 CE, when the astronomer and mathematician Aryabhata (476–550) used infinitesimals to study the motion of the moon.The use of infinitesimals to compute rates of change was developed significantly by Bhāskara II (1114-1185); indeed, it has been argued[3] that many of the key notions of differential calculus can be found in his work, such as "Rolle's theorem".The Persian mathematician, Sharaf al-Dīn al-Tūsī (1135-1213), was the first to discover the derivative of cubic polynomials, an important result in differential calculus;[5] his Treatise on Equations developed concepts related to differential calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions. An early version of the mean value theorem was first described by Parameshvara (1370–1460) from the Kerala school of astronomy and mathematics in his commentary on Bhaskara II.
The modern development of calculus is usually credited to Isaac Newton (1643 – 1727) and Gottfried Leibniz (1646 – 1716), who provided independent and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes, which had not been significantly extended since the time of Ibn al-Haytham (Alhazen). For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as Isaac Barrow (1630 – 1677), René Descartes (1596 – 1650), Christiaan Huygens (1629 – 1695), Blaise Pascal (1623 – 1662) and John Wallis (1616 – 1703). In particular, Isaac Barrow is often credited with the early development of the derivative. Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the notation still used today.
Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy (1789 – 1857), Bernhard Riemann (1826 – 1866), and Karl Weierstrass (1815 – 1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.
DERIVATIVES RULES:
1. Constant Rule:
If y = k, then y' = 0
The Derivative of a Constant is 0
If ƒ(x) =k for some constant k, then ƒ'(x) = 0
2. Power Rule
If y = x", then y' = nxn-1
If ƒ is a differentiable function, and if ƒ(x) = x", then ƒ'(x) = nxn-1 for any real number n
3. Exponential Rule:
If y = ex, then y' = ex
4. Logarithm Rule:
If y = 1n|x|, then y' = 1/x
5. Constant Times a Function Rule:
If y = kƒ, then y' =kf '
6. Sum Rule
If y = ƒ g, then y' = ƒ' g'
7. Product Rule
If y = ƒg, then y' = ƒg' ƒ' g
If ƒ and g are differentiable functions such that y = ƒ(x)g(x), then y' = ƒ(x)g' ƒ' (x)g(x)
8. Difference Rule
If y = ƒ - g, then y' = ƒ' - g'
9. Quotient Rule
To remember this formula: Simply remember that b comes before t in the alphabet Thus, the bottom function times the derivative of the top minus the top times the derivative of the bottom, all divided by the bottom squared!
10. Chain Rule:
If y is a differentiable function of u and u is a differentiable function of x and
The Derivative
Definition of The Derivative
The derivative of the function f(x) at the point is given and denoted by
Some Basic Derivatives
In the table below, u,v, and w are functions of the variable x. a, b, c, and n are constants (with some restrictions whenever they apply). designate the natural logarithmic function and e the natural base for . Recall that .
Chain Rule
The last formula
is known as the Chain Rule formula. It may be rewritten as
Another similar formula is given by
Derivative of the Inverse Function
The inverse of the function y(x) is the function x(y), we have
Derivative of Trigonometric Functions and their Inverses
Recall the definitions of the trigonometric functions
Derivative of the Exponential and Logarithmic functions
Recall the definition of the logarithm function with base a > 0 (with ):
Derivative of the Hyperbolic functions and their Inverses
Recall the definitions of the trigonometric functions
Higher Order Derivatives
Let y = f(x). We have:
In some books, the following notation for higher derivatives is also used:
Higher Derivative Formula for the Product: Leibniz Formula
where are the binomial coefficients. For example, we have
is an engineering discipline which uses the scientific knowledge of the behavior and effects of electrons to develop components, devices, systems, or equipment (as in electron tubes, transistors, integrated circuits, and printed circuit boards) that uses electricity as part of its driving force. Both terms denote a broad engineering field that encompasses many sub fields including those that deal with power, instrumentation engineering, telecommunications, semiconductor circuit design, and many others.
is given and denoted by 
designate the natural logarithmic function and e the natural base for 
.
): 
are the binomial coefficients. For example, we have 
