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[PDF] THE DERIVATIVE
Here, we represent the derivative of a function by a prime symbol For example, writing ? represents the derivative of the function evaluated at point
[PDF] AN INTRODUCTION TO DERIVATIVE MARKETS
Examples of underlying assets are stocks, bonds, and commodities There are four main reasons for the use of derivatives The first is risk management When
[PDF] Definition of derivative
Definition of Derivative • 6 Example • 7 Extension of the idea • 8 Example • 9 Derivative as a Function • 10 Rules of Differentiation
[PDF] Rules for Finding Derivatives
Fortunately, we can develop a small collection of examples and rules that allow us to compute the derivative of almost any function we are likely to
[PDF] BASICS OF EQUITY DERIVATIVES - BSE
Such a transaction is an example of a derivative The price of In the class of equity derivatives the world over, futures and options on stock
[PDF] Derivatives markets, products and participants: an overview
For example, banks often offer more favourable financing terms to those firms that have reduced their market risks through hedging activities than to those
[PDF] FINANCIAL DERIVATIVES CLASSIFICATION OF DERIVATIVES
A derivative is a financial instrument or contract that derives its value from an underlying asset For example, a wheat farmer may wish to contract to
[PDF] Financial Derivatives - International Monetary Fund
Financial derivatives contracts are usually settled by net payments of cash, often For example, the counter party to a derivative contract that
[PDF] Derivative Securities - University of Scranton
The main types of derivatives are futures, forwards, options, and swaps An example of a derivative security is a convertible bond Such a bond, at the
[PDF] FINANCIAL DERIVATIVES - IARE
For example, gold is widely considered a good investment to hedge against stocks and currencies When the stock market as a whole isn't performing well, or
[PDF] BASICS OF EQUITY DERIVATIVES - BSE
It does not own any shares or index futures or commodities which might be traded on the exchange For example, an exchange where gold futures are traded
[PDF] 25 Summary of derivative rules
2 Example Find the derivative of f(x) = 3 / tan 5x + 1 Page 6 Summary of derivative rules Tables Examples
[PDF] Financial Derivatives - International Monetary Fund
Examples are bonds that are convertible into shares and securities that carry the option of repaying the principal in a different currency from that of issuance
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832_2derivsummary_screen.pdf Summary of derivative rules
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JJII JI Page 1 of 11 BackPrint Version
Home Page25.Summary of deriv ativerules
25.1.T ables
The derivative rules that have been presented in the last several sections are collected together in the following tables. The rst table gives the derivatives of the basic functions; the second table gives the rules that express a derivative of a function in terms of the derivatives of its component parts (the \derivative decomposition rules"). For the sake of completeness, a few new rules have been added.Summary of derivative rules
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Home PageDerivatives of basic functions.
ddx [c] = 0 (c, constant)ddx [xn] =nxn 1 ddx [ex] =exddx [ax] =axlna ddx [lnjxj] =1x ddx [logajxj] =1xlna ddx [sinx] = cosxddx [cosx] = sinx ddx [tanx] = sec2xddx [cotx] = csc2x ddx [secx] = secxtanxddx [cscx] = cscxcotx ddx sin 1x=1p1 x2ddx cos 1x= 1p1 x2 ddx tan 1x=11 +x2ddx cot 1x= 11 +x2 ddx sec 1x=1x px2 1ddx
csc 1x= 1x px 2 1Summary of derivative rules
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Home PageThe rules for the derivative of a logarithm have been extended to handle the case ofx <0 by the addition of absolute value signs. If the absolute value signs are removed, the rules are still valid, but only forx >0.Summary of derivative rules
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Home PageDerivative decomposition rules.
Constant multiple rule
ddx [cf(x)] =cddx [f(x)]Sum/Di erence rule
ddx [f(x)g(x)] =ddx [f(x)]ddx [g(x)]Product rule
ddx [f(x)g(x)] =ddx [f(x)]g(x) +f(x)ddx [g(x)]Quotient rule
ddx f(x)g(x) =g(x)ddx [f(x)] f(x)ddx [g(x)](g(x))2Chain rule
ddx [f(g(x))] =f0(g(x))g0(x)Summary of derivative rules
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Home Page25.2.Examples
25.2.1 ExampleVerify the ruleddx
[tanx] = sec2x.Solution
ddx [tanx] =ddx sinxcosx = cosxddx [sinx] sinxddx [cosx](cosx)2 = cosx(cosx) sinx( sinx)cos 2x = cos2x+ sin2xcos 2x = 1cos2x= sec2x:25.2.2 ExampleFind the derivative off(x) =3ptan5x+ 1.
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Home PageSolution
f0(x) =ddx
3ptan5x+ 1
= 13 (tan5x+ 1) 2=3ddx [tan5x+ 1] = 13 (tan5x+ 1) 2=3sec25x5 =5sec25x3(
3ptan5x+ 1)2:25.2.3 ExampleFind the derivatives of each of the following functions. Avoid the
quotient rule. (a)f(x) =sin3x4 (b)f(x) =4sin3x SolutionWhen either the numerator or the denominator is constant, the quotient rule can be avoided by rst rewriting, and such a solution is generally easier than one using the quotient rule. (If neither numerator nor denominator is constant, then rewriting in order to use the product rule requires more steps than just using the quotient rule so is not advised.) (a)Rewriting as f(x) =14
sin3x, we havef0(x) =14 cos3x3.Summary of derivative rules
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JJII JI Page 7 of 11 BackPrint Version
Home Page(b)Rewriting as f(x) = 4(sin3x) 1, we have f0(x) = 4(sin3x) 2cos3x3 = 12cos3xsin
23x:25.2.4 ExampleFind the derivative ofy=7tan
12x. SolutionWe rst rewrite asy= 7(tan 12x) 1to avoid using the quotient rule. (Inciden- tally, the 1's do not cancel since tan 12xdenotes the inverse tangent of 2xrather than (tan2x) 1.) We have y0= 7(tan 12x) 211 + (2x)22 = 14(tan
12x)2(1 + 4x2):25.2.5 ExampleDi erentiatef(t) =t3esec2t.Solution
f0(t) =ddt
t3esec2t = ddt t3esec2t+t3ddt esec2t = 3t2esec2t+t3esec2tddt [sec2t] = 3t2esec2t+t3esec2tsec2ttan2t2 =esec2t 3t2+ 2t3sec2ttan2t:Summary of derivative rules
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JJII JI Page 8 of 11 BackPrint Version
Home Page25.2.6 ExampleFind the derivative off(x) = cotxx. SolutionSince cotxxmeans (cotx)x, this is a case where neither base nor exponent is constant, so logarithmic di erentiation is required: ln(f(x)) = lncotxx =xlncotx; so ddx [ln(f(x))] =ddx [xlncotx]1f(x)f0(x) =ddx
[x]lncotx+xddx [lncotx] = lncotx+x1cotxddx [cotx] = lncotx+x1cotx( csc2x) = lncotx xcosxsinx; and nally f0(x) =f(x)
lncotx xcosxsinx = cot xx lncotx xcosxsinx :Summary of derivative rules
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Home Page25.2.7 ExampleGiveny= csc352x+xxsin3x ln(4 +x2) , ndy0without using inter- mediate steps.Solution
y0= 3csc252x+xxsin3x ln(4 +x2)
csc52x+xxsin3x ln(4 +x2) cot52x+xxsin3x ln(4 +x2) (xsin3x ln(4+x2))(52xln5(2)+1) (52x+x) sin3x+3xcos3x 2x4+x2(xsin3x ln(4+x2))2: (We used, in order, (1) the chain rule with the cubing function as the outside function, (2) the chain rule with the cosecant function as the outside function, (3) the quotient rule.)Summary of derivative rules
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JJII JI Page 10 of 11 BackPrint Version
Home Page25{Exercises
25{1V erifythe rule ddx [secx] = secxtanx. 25
{2Find the deriv ativeof eac hof the follo wingfunctions: (a)f(x) =5p4x sec3x. (b)f(t) = tan 1e5t. 25
{3Find the deriv ativeof eac hof the follo wingfunctions: (a)y= 2csc7xtan(ln4x). (b)y= sec 1e3t1 +t2 . 25
{4Di eren tiatef(x) = tanexx.