2 Objective MHT-CET Mathematics 2 Differentiation

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2Objective MHT-CET Mathematics2Differentiation2.1Relationship BetweenDifferentiabilityContinuity2.2Derivative of Composite Functions2.3Derivative of Inverse Functionsand2.4Logarithmic Differentiation2.5Derivative of Implicit Functions2.6Derivative of Parametric Functions2.7Higher Order DerivativesIntroduction Let y f (x) be a real valued function, then derivative ofthe function w.r.t. x is given bydyf (x h) f (x )f ′(x) or limdx h 0hdyThe derivative f ′(x) oris also known as differentialdxcoefficient of f (x).If the derivative of a function at a point exists thenthe function is called differentiable or derivable at thatpoint.Illustration: If f (x) x sin x, find f ′(p), using firstprinciple.Soln.: f (x) x sin x\ f (p) (p) sin p 0f (p h) (p h)sin(p h)By first principlef ( p h) f ( p)f ′( p) limhh 0(p h)sin(p h) 0 limhh 0( p h)(sin p cos h cos p sin h) limhh 0( p h)sin h lim –(p 0) 1 – phh 0Illustration: If f (x) log(3x – 1), find f ′(2), using firstprinciple.Soln.: Given f (x) log (3x – 1)\ f (2) log (6 – 1) log 5and f (2 h) log [3(2 h)–1] log (3h 5)By first principlef ′(2) limh 0f (2 h) f (2)hlog(3h 5) log 5hh 0 lim13h 5 13hlog lim log 1 hh 0 5h 0 h5 lim3hhlog 1 log 1 3 3 3 55 lim lim5 5 h 0h 0 3h 3h 5 53hlog(1 t ) 3 1 5 1 h 0, 5 0 and tlimt 03 5Left Hand Derivative If y f (x) is a real valued function and a is any realf (a h ) f (a )number, then lim, if it exists, is hh 0called the left hand derivative of f (x) at x a and isdenoted by Lf ′(a) or f ′(a) – or f ′(a–).Right Hand Derivative If y f (x) is a real valued function and a is any realf (a h ) f (a )number, then lim, if it exists, is hh 0called the right hand derivative of f (x) at x a and isdenoted by Rf ′(a) or f ′(a) or f ′(a ).Note : If f ′(a ) and f ′(a–) are equal then f (x) is said tobe differentiable at x a.

Differentiation3Illustration: Discuss the differentiability of f(x) x x at x 0. x 2, x 0 Soln.: We have, f (x) x x 2 x , x 0f (x) f (0)Now, (LHD at x 0) lim x 0x 0 x 2 0x 0 x 0 (LHD at x 0) lim x 0 (LHD at x 0) limf (x) f (0)and, (RHD at x 0) limx 0x 0 2x 0 (RHD at x 0) limx 0 x 0 (RHD at x 0) lim x 0x 0So, f (x) is differentiable at x 0.2.1 Relationship Between Continuityand Differentiability Every differentiable function is continuous but theconverse need not be true.Illustration: A function is defined byp 1, for 2 x 0f (x) p 1 sin x, for 0 x .2 What can you say about right hand derivative and lefthand derivative at x 0?Is f continuous at x 0 ?Soln.: Right hand derivative at x 0 isf (0 h) f (0)f ′(0 ) lim hh 0f (h) f (0) lim hh 0(1 sin h) (1 sin 0) limhh 0p [. f (x) 1 sinx, 0 x ]2sin h1 sin h 1 lim lim 1hh 0h 0 hLeft hand derivative at x 0 isf (h) f (0)f (0 h) f (0)f ′(0 ) lim lim hhh 0h 01 (1 sin 0)1 1 00 lim lim lim 0hhh 0h 0h 0 h Since f ′(0 ) f ′(0 –)\ f is not differentiable at x 0.Continuity at x 0\(h 0, h 0)x 0 x 0f (x) 1 sinx, for 0 x p2\ f (0) 1 sin 0 1and lim f (x) lim(1 sin x ) 1 sin0 1\x 0p x 02lim f (x) lim(1) 1f (x) 1, for \x 0 x 0lim f (x) f (0) lim f (x)x 0 x 0 f is continuous at x 0.2.2 Derivative of Composite Functions If y f (u) is a differentiable function of u and u g(x)is a differentiable function of x, then y f (g(x)) is ady dy du differentiable function of x anddx du dxThis is called chain rule. If y is a differentiable function of u1, ui is a differentiablefunction of ui 1, for i 1, 2, . n – 1 and un is adifferentiable function of x, then y is a differentiablefunction of x anddudy dy du1 du2 . ndx du1 du2 du3dxIllustration: Differentiate : sin (x2 5) w.r.t. xSoln. : Let y sin (x2 5) (i)Also let u x2 5 \y sinu (ii)dyThen,(from (ii)) cos u dudu 2x (from (i))dxdy dy du cos(x 2 5) 2xBy chain rule,dx du dx 2x cos (x2 5)3Illustration: If y (7x 2 11x 39) 2 , findSoln.: y (7x2 11x 39)3/2Let u 7x2 11x 39 y u3/2 \13dy 3 2 1 3 2 u (u)2du 2dy.dx (i) (ii)(from (ii))du 14x 11 (from (i))dxSo, by chain ruledy dy du 3 2 (7x 11x 39)1/ 2 (14x 11)dx du dx 23 (14x 11) 7x 2 11x 392

4Objective MHT-CET MathematicsDerivative of Some Standard CompositeFunctionsSr. No.1.[ f (x)]n2.f (x)3.dydxy1f (x)2.3.1 Derivative of Inverse TrigonometricFunctionsSr. No.2 f (x) 1[ f (x)]2f ′(x)11.sin–1x, x [–1, 1] f ′(x)2.cos–1x, x [–1, 1] f ′(x)3.tan–1x, x R4.cot–1x, x R5.sec–1x,x (– , –1) (1, )6.cosec–1x,x (– , –1) (1, )n [ f (x)]n–1 f ′(x)1f (x)4.sin[ f (x)]cos[ f (x)] f ′(x)5.cos[ f (x)]– sin[ f (x)] f ′(x)6.sec[ f (x)]sec[ f (x)] · tan[ f (x)] f ′(x)7.cosec[ f (x)]– cosec[ f (x)] · cot[ f (x)] f ′(x)8.tan[ f (x)]sec2[ f (x)] f ′(x)9.cot[ f (x)]– cosec2[ f (x)] f ′(x)10.a f (x)a f (x) · log a f ′(x)11.e f (x)12.log[ f (x)]13.loga[ f (x)]e f (x) f ′(x)1 f ′(x)f (x)1 f ′(x)f (x) log e aIllustration: Differentiate : sin[cos (tan x)] w.r.t. xSoln. : Let y sin[cos (tan x)]dy d\ {sin[cos(tan x)]}dx dxd cos[cos(tan x )] [cos(tan x)]dxd cos[cos(tan x )] [ sin(tan x)] (tan x)dx – cos[cos(tanx)] · sin(tanx) · sec2x – sec2x · cos[cos(tanx)] · sin(tan x).2.3 Derivative of Inverse Functions If y f (x) is a differentiable function of x such thatthe inverse function x f –1(y) exists, then x is adifferentiable function of y anddydx1 , where 0dy dydxdx1 x2 11 x21, x 1, x 11 x2 11 x212x x 1 12x x 1, x 1, x 1 1x w.r.t. xIllustration: Differentiate : cotSoln. : Let y cot 1 xdy d 1d (cot 1 x ) x2 dxdx dx1 ( x) 111 1 x 2 x2 x (1 x)2.3.2 Derivative of Inverse CompositeFunctionsFunctionsin-1( f (x))cos–1( f (x))tan–1( f (x))cot–1( f (x))sec–1( f (x))cosec–1( f (x))Derivative11 ( f (x))2 11 ( f (x))211 ( f (x))2 11 ( f (x))2 f ′(x), f (x) 1 f ′(x), f (x) 1 f ′(x) f ′(x)1f (x) ( f (x))2 1 1f (x) ( f (x))2 1 f ′(x), f (x) 1 f ′(x), f (x) 1

Differentiation5Illustration: Differentiate : sin–1(cos3x) w.r.t. xSoln.: Let y sin–1(cos3x)dy d\ [sin 1(cos 3x)]dx dxd1 (cos 3x)dx21 (cos 3x)1 ( sin 3x) d(3x)dx1 cos 2 3x11 ( sin 3x) 3 ( sin 3x) 3 3sin 3xsin 2 3xIllustration: Differentiate : 1 sin x 1 sin x tan 1 w.r.t. x 1 sin x 1 sin x 1 1 sin x 1 sin x Soln.: Let y tan 1 sin x 1 sin x Now,1 sin x cos 2Similarly,xxxx sin 2 2 sin cos2222xx cos sin22xx1 sin x cos sin22xx xx cos sin cos sin 2222y tan xxxx cos sin cos sin 22 22 x 2 cosx 1 2 tan 1 tan cot 2 x 2 sin 2 \ 1 p x p x tan 1 tan 2 2 2 2 dy1 Diff. w.r.t. x, we getdx2Derivatives of InverseFunctions by SubstitutionTrigonometric With proper substitution the given inverse functioncan be reduced to a simple form. Then the derivativemay be easily obtained.Standard substitutions are :Sr. Expression involving SubstitutionNo. the term1.a2 x2x a sin q or a cos q2.a2 x2x a tan q or a cot q3.x2 a2x a sec q or a cosec q4.5.6.7.a x or a xx a cos q or a cos2q1 – 2x2x sin q2x2–12x2,x cos q2x1 x 1 x21 x2x tanqand1 x2Illustration: Differentiate : 1 x 1 x tan 1 w.r.t.x 1 x 1 x 1 1 x 1 x Soln.: Let y tan 1 x 1 x 1Put x cos 2q q cos 1 x2 cos12q 1 cos 2q \ y tan 1 1 cos 2q 1 cos 2q 2 cos 2 q 2 sin 2 q tan 1 2 cos 2 q 2 sin 2 q qq tan 1 cos sin cos q sin q 1q tan 1 tan tan 1 tan p q 1 tan q 4pp 1 q cos 1 x44 2On differentiating w.r.t. x, we getdy1 1 1 2dx2 1 x 2 1 x2 x ax 1 aIllustration: Differentiate : cos xw.r.t.x a a x 1 ax xx x a a 1 cos 1 aSoln.: Let y cos x x1x a a a ax2x 1 a cos 1 1 a 2x Put ax tanq q tan–1(ax) 1 tan 2 q \ y cos 1 cos 1(cos 2q) 1 tan 2 q 2 q 2 tan–1(ax)dydd1\ 2 [tan 1(a x )] 2 (a x )x 2 dxdxdx1 (a ) 21 a 2x a x log a 2a x log a1 a 2x

6Objective MHT-CET Mathematics2.4 Logarithmic Differentiation When we want to find the derivative of a functionwhich can be expressed as :(i) a product of number of functions(ii) a quotient of functions(iii) a function which is of the form [f (x)]g(x),then it is easy to find the derivative of the logarithm ofthe function.This method is known as logarithmic differentiation.Note that by chain rule,dy 1 dydddx (log y) dy (log y). dx y dx . y x 2 dyIf x exp tan 1 , find dx .2 x 1 y x 2 Soln.: x exp tan x 2 Taking log on both sides, we get y x2 log x tan 1 x2 y x2 tan(log x) y x2 tan(logx) x2x2On differentiating w.r.t. x, we getsec2(log x)dy 2x tan(log x) x 2 2xdxxdy 2x tan(log x) x sec2(log x) 2xdxdy 2x[1 tan(log x)] x sec2(log x)dxIllustration:2.5 Derivative of Implicit Functions If a relation between x and y is such that y can beexpressed in terms of x i.e., y f (x) then y is calledexplicit function of x. If a relation between x and y is such that y cannot beexpressed in terms of x, then y is called an implicitfunction of x. When a given relation expresses y as an implicitdy, then wefunction of x and we want to finddxdifferentiate every term of the relation w.r.t. x,remembering that a term in y is first differentiated w.r.t.dyy and then multiplied by.dxNote : While taking the derivative of implicit function,function of y is considered as composite function of x.dyIllustration: If xmyn (x y)m n, then finddxSoln.: Given xmyn (x y)m nTaking log on both sides, we getm log x n log y (m n) log(x y)Differentiating both sides w.r.t. x, we get m n dy m n dy 1 x y dx x y dx m m n m n n dy x x y x y y dx my nx my nx dydy y x(x y) y(x y) dxdx xIllustration: x y a, then find dy .If sec x y dxx y x y sec 1 a a Soln.: Given sec xy xy Differentiating both sides w.r.t. x, we getdydy(x y) 1 (x y) 1 dx dx 0(x y)2dydy (x y ) (x y ) 0dxdxdydydy y 2 y (x y x y) 2 y 2x dxdxdx x x y (x y )2.6 Derivative of Parametric Functions If x and y are expressed as functions of the variable t,i.e., x f (t) and y g(t), then these equations are calledparametric equations and t is called parameter. If x f (t) and y g(t) are differentiable functions ofparameter t, then y is a differentiable function of x anddydy dt dx , 0dx dx dtdt2t1 t2, then findIllustration: If x and y 21 t21 tdy.dxSoln. : x 1 t2and y 2t1 t1 t2Put t tanq in both the equations, we getx and y 1 tan 2 q1 tan 2 q2 tan q2 cos 2q sin 2q1 tan 2 qOn differentiating (1) and (2) w.r.t. q, we getdydx 2 sin 2q and 2 cos 2qdqdqdydy d qxcos 2qTherefore, dx dxysin 2qdq (1) (2)

Differentiation7Illustration:y cos–1If x sin–1(3t – 4t3) anddy( 1 t 2 ), then find.dxSoln. : Given that, y cos 1 1 t 2 sin 1 t . (1). (2)and x sin–1(3t – 4t3) 3 sin–1t On differentiating (1) and (2) w.r.t. t, we getdy1dx3 anddtdt21 t1 t2 1 dy dy dt1 t2 1 dx dx 1 3 dt 3 1 t2 \2 dy .(x 2 a 2 ) 1dxAgain, differentiating w.r.t. x, we get2.7 Higher Order Derivatives (SecondOrder Derivatives) Derivative of y f (x) w.r.t. x (if it exists) is denoted bydyor f ′(x) and is called the first order derivative of y.dxdyDerivative ofor f ′(x) w.r.t. x (if it exists) is denoteddx2d yby 2 or f ′′(x) and is called the second order derivativedxof y.Derivative ofd2ydx 2or f ′′(x) w.r.t. x (if it exists) is3denoted by d y or f ′′′(x) and is called the third orderdx 3derivative of y.dnyis the nth order derivative of y w.r.t. x.dx nNote : These higher order derivatives may also bedenoted by y1, y2, y3, . ynIn general,Illustration: If y x3 log x, then findd2 ydx 2.Soln. : Let y x3 log xdy1\ x 3 3x 2 log x x 2 3x 2 log xdxxAgain differentiating w.r.t. x, we getd2ydx2 2x 3x 2 1 6x log x 5x 6x log xx22Illustration: If y log{x x a }, then prove that2dyd y(x 2 a 2 ) 2 x 0.dxdxSoln. : We have, y log{x x 2 a 2 }On differentiating w.r.t. x, we getdy1d (x x 2 a 2 )dx x x 2 a 2 dx2x 1 1 22 x x2 a2 2 x a x 2 a 2 x 1 x x 2 a 2 x 2 a 2 1 x2 a222 dy d (x 2 a 2 ) (x 2 a 2 ) d dy 0 dx dxdx dx 2dy 2 dy 2x (x 2 a 2 ) 2 d y 0dxdx dx 2 2dy d 2 y 2 2dy 2 (x a ) x 0dx dxdx (x 2 a 2 )d2 ydx2 xdy 0 dx[Q y1 0]Illustration: If y log(1 cos x), then prove thatd3yd 2 y dy 0dx 3 dx 2 dxSoln. : Given, y log(1 cos x)dy1 sin x( sin x) dx 1 cos x1 cos x (1 cos x) cos x sin x ( sin x) dx 2(1 cos x)2 cos x cos 2 x sin 2 x (1 cos x)2 1 cos x 1 2 1 cos x (1 cos x) d 2yd 3y ( 1)( 1)(1 cos x) 2( sin x)dx sin x (1 cos x)2 3Now,d3ydx 3 d 2 y dy dx 2 dx sin x 1 sin x (1 cos x) 1 cos x 1 cos x 2 sin x sin x(1 cos x)2 0

Differentiation7Multiple Choice QuestionsLEVEL - 12.1 Relationship Between Continuityand Differentiability1.2.3.4.5.6.7. x sin(1 / x), when x 0The function f (x) , iswhen x 0 0,(a) continuous but not differentiable at x 0(b) continuous and differentiable at x 0(c) neither continuous nor differentiable at x 0(d) none of thesef(x) x3 is(a) continuous but not differentiable at x 3(b) continuous and differentiable at x 3(c) neither continuous nor differentiable at x 3(d) none of thesef(x) [x] is neither continuous nor differentiable at(a) integral points(b) rational points(c) real points(d) none of thesef(x) x – 2 is(a) continuous but not differentiable at x 2(b) continuous and differentiable at x 2(c) neither continuous nor differentiable at x 2(d) none of these (2 x), when x 1, then f(x) isLet f (x) x, when 0 x 1(a) continuous but not differentiable at x 1(b) continuous and differentiable at x 1(c) neither continuous nor differentiable at x 1(d) none of these (1 x), when x 1The function f (x) 2is (x 1), when x 1(a) continuous but not differentiable at x 1(b) continuous and differentiable at x 1(c) neither continuous nor differentiable at x 1(d) none of these 1 x, for x 2, thenIf f (x) 5 x, for x 2(a) f(x) is continuous and differentiable at x 2(b) f(x) is continuous but not differentiable at x 2(c) f(x) is everywhere differentiable(d) f(x) is not continuous at x 28. 1 x cos , for x 0, then at x 0, f(x) isIf f (x) x 0,for x 0(a)(b)(c)(d)9.not continuouscontinuous and differentiablecontinuous but not differentiablenone of these x 1, for x 2If f (x) , then at x 2, f(x) is 2x 3, for x 2(a) continuous but not differentiable(b) continuous and differentiable(c) neither differentiable nor continuous(d) none of these p 1 x sin , for x 0, then f(x) isx10. Let f (x) 0,for x 0continuous but not differentiable at x 0, if(a) 1 p (b) 0 p 1(c) – p 0(d) p 011. If f(x) [x], – 2 x 2, then at x 1(a) f(x) is continuous and differentiable(b) f(x) is neither continuous nor differentiable(c) f(x) is continuous but not differentiable(d) none of these x, for x 012. If f (x) , then at x 0 0, for x 0(a) f(x) is differentiable and continuous(b) f(x) is neither continuous nor differentiable(c) f(x) is continuous but not differentiable(d) none of these13. Which of the following is not always true?(a) If f(x) is continuous at x a, then it isdifferentiable at x a(b) If f(x) is not continuous at x a, then it is notdifferentiable at x a(c) If f(x) and g(x) are differentiable at x a, thenf(x) g(x) is also differentiable at x a(d) If f(x) is continuous at x a, then lim f (x)x aexists.

8Objective MHT-CET Mathematics14. If f(x) x – 1 , x R, then at x 1(a) f(x) is not continuous(b) f(x) is continuous but not differentiable(c) f(x) is continuous and differentiable(d) none of these x, for 0 x 1, then15. If f (x) 2x 1, for 1 x(a) f(x) is discontinuous at x 1(b) f(x) is differentiable at x 1(c) f(x) is continuous but not differentiable at x 1(d) none of these 1 16. Let f (x) (x a)cos for x a and let f(a) 0.x a Then f(x) at x a is(a) continuous but not differentiable(b) continuous and differentiable(c) neither continuous nor differentiable(d) none of these17. 1 x, x 0f (x) at x 0 is 1 x, x 0(a) discontinuous(b) continuous but not differentiable(c) differentiable(d) none of these18. Find the values of a, b respectively if2 x 3x a, x 1 is differentiable at every x.f (x) x 1 bx 2,(a) 3, 2(b) 2, 4(c) 3, 5(d) 5, 319. Let f(x) x x . Then f at x 0 is(a) continuous and differentiable(b) continuous but not differentiable(c) neither continuous nor differentiable(d) none of these20. Examine the differentiability of the function f defined 2x 3, if 3 x 2 by f (x) x 1, if 2 x 0 x 2, if 0 x 1 (a) differentiable at all x R –{0, – 2}.(b) continuous only(c) discontinuous everywhere(d) differentiable at all x Rx 1 x 3 , 21. The function f (x ) x 2 3is13 x , x 1 4 24(a) continuous and differentiable at x 1(b) continuous but not differentiable at x 1(c) discontinuous at x 1(d) none of these22.f(x) e x is(a) discontinuous everywhere(b) differentiable at x 0(c) continuous at x 0 only(d) not differentiable at x 023. The function f(x) e– x is(a) continuous everywhere but not differentiable atx 0(b) continuous and differentiable everywhere(c) not continuous at x 0(d) none of these24. Let f(x) sin x . Then,(a) f(x) is everywhere differentiable(b) f(x) is everywhere continuousdifferentiable at x n p, n Z(c) f(x) is everywhere continuouspdifferentiable at x (2n 1) , n Z2(d) none of thesebutnotbutnot25. The function f(x) cos x is(a) differentiable at x (2n 1) p/2, n Z(b) continuous everywhere but not differentiable atx (2n 1) p/2, n Z(c) neither differentiable nor continuous at x np,n Z(d) none of these26. If f (x) 3 – x (3 x), where (x) denotes the leastinteger greater than or equal to x, then f(x) is(a) continuous and differentiable at x 3(b) continuous but not differentiable at x 3(c) neither differentiable nor continuous at x 3(d) none of these27.Let(a)(b)(c)(d)x 1 1, f (x) x , 1 x 1 . Then, f is 0,x 1 continuous at x –1differentiable at x – 1everywhere differentiablenone of these28. f(x) x3 at x 0 is(a) continuous and differentiable(b) continuous but not differentiable(c) discontinuous(d) none of these

Differentiation x 2 2x, x 0, then value of a and b such29. If f (x) ax b, x 0that f(x) is continuous and differentiable at x 0 are(a) a 1, b 0(b) a 2, b 0(c) a 0, b 2(d) a 0, b 13 3 2x, 2 x 0, then f(x) at x 0 is30. If f (x) 3 2x, 0 x 3 2(a) discontinuous(b) differentiable(c) continuous but not differentiable(d) none of these31. Values of a and b such that the function x 2 if x 1is derivable at x 1 aref (x) 2ax b if x 1respectively(a) 1, –1(b) –1, 1(c) 1, 1(d) –1, –132. If f(x) [x], where [x] denotes the greatest integer lessthan or equal to x. Then, f(x) at x 0 is(a) neither continuous nor differentiable(b) continuous only(c) differentiable(d) none of these9(a)(b)(c)(d)37.35. f(x) 1 x, if x 2is 5 – x, if x 2,(a) discontinuous at x 2(b) continuous but not differentiable at x 2(c) continuous and differentiable at x

2 Objective MHT-CET Mathematics Introduction Let y f (x) be a real valued function, then derivative of the function w.r.t. x is given by fx dy dx fxhfx h h ′( )lim ()() or 0 hThe derivative fx dy dx ′( )or is also known as differential coefficient of (x)

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