Differentiation And Its Application In Business 5u3m1g

Differentiation and its Application in Business.  ¾ What is Differnetiation ?  ¾ Definition.  ¾ Historical BackGround.  ¾ Notations Used to Represent the Differentiation.  ¾ Rules and Formulas  ¾ Computing The Derivatie (An Example)  ¾ Applications of Differnetiation.  ¾ Case Study   

De efinition   D Derivative e is a meaasure of h how a function changes as itts input ch hanges. Lo oosely  s speaking,  a derivattive can be thoughtt of as how much o one quanttity is changing  i in respon se to changes in so ome other quantityy. 

  unction, d drawn in b black, and d a tangen nt line to tthat functtion,  The graaph of a fu drawn in red. Th he slope o of the tanggent line is equal to the derivative off the  fun nction at tthe markeed point.  

Historical BackGround  ¾ 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).  ¾ 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.  ¾ The use of infinitesimals to compute rates of change was developed significantly by Bhāskara II (1114‐1185)  ¾ 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  ¾ The modern development of calculus is usually credited to Isaac Newton (1643 – 1727) and Gottfried  Leibniz (1646 – 1716). 

N Notations  Used to Represen nt the Diffferentiation  1 1.Leibniz z's notatio on F First deriv vative is denoted d by y

Higher deerivatives are expreessed H u using the notation

chaiin rule eassy to remeember

2 Lagran 2. nge's nota ation Sometimees referred S d to as prrime notationone of o the mosst common modern n n notations for differrentiation..  

and d 

To denotee the num T mber of derrivatives beyond b th his point, some s authhors use Roman R n numerals in superscript, wheereas otheers place the t numbeer in parenntheses:  

or  

3 Newton 3. n's notatiion Newton's notation for differeentiation, also calleed the dott notation,, places a dot N o over the function f n name to reepresent a time deriivative. Iff y = ƒ(t), then  

and a  

4 Euler'ss notation 4. n Euler's no E otation usees a differrential operator D, which is applied too a functio on ƒ to 2 g the first give f derivaative Df. The T secon nd derivattive is den noted D ƒƒ, and the nth d derivative e is denoteed Dnƒ. IIf y = ƒ(x)) is a depeendent vaariable, theen often th he subscript x is atttached to the D t clarify the indep to pendent vaariable x. Euler's no otation is then writtten      

or  

,

Formulas and Ru ules:  Deriivatives of poweers: if where r is any a real numberr, then

E Exponential and d logarithmic fu unctions:

•

•

Trigo onometric functtions:

Inversee trigonometric functions: f

Rulees for finding the derivativ ve S Some of the most basic rules arre the following. •

•

Quotient rule:

Constant rule: if ƒ(x) is constant, then for alll functions ƒ and d g where

•

g ≠ 0.

Sum rulee: for all funcctions ƒ and g

•

Chain rule: If f(x) f

= h(g(x))), then

and all reeal numbers a and a b. Product rule: fo all functions ƒ and g. for

E Example :

F Find The deerivative of

?

t second term m was computeed using the cha ain rule and thiird using the prroduct rule. Th he known deriv vatives of Here the the elementa ary functions x2, x4, sin(x), ln(x x) and exp(x) = ex, as well as the t constant 7, were also used d.