METHODS OF DIFFERENTIATION Contents Topic Page No. Theor 0 0 Objective Question 0 09 Subjective Question 0 NCERT Board Questions Answer Ke 4 Sllabus Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polnomial, rational, trigonometric, inverse trigonometric, eponential and logarithmic functions. Derivatives of implicit functions, derivatives up to order two, Name : Contact No. ARRIDE LEARNING ONLINE ELEARNING ACADEMY A479 indra Vihar, Kota Rajasthan 4005 Contact No. 80545007
. DEFINITION : METHODS OF DIFFERENTIATION KEY CONCEPTS If and + h belong to the domain of a function f defined b = f (), then f( + h) Lim 0 h h f() if it eists, is called the DERIVATIVE of f at & is denoted b f () or. we have therefore d D [f()] = f' () = d = () = Lim h 0 f ( + h) f() h f (a + h) f(a). The derivative of a given function f at a point = a of its domain is defined as Lim, h 0 h provided the limit eists & is denoted b f (a). Note that alternativel, we can define f' (a) = Lim f() f(a), provided the limit eists a a. DERIVATIVE OF f () FROM THE FIRST PRINCIPLE / ab INITIO METHOD / DELTA METHOD: f ( + d) f () If f () is a derivable function then, Lim = Lim = f' () = d 0 d d 0 d d 4. THEOREMS ON DERIVATIVES : If f and g are derivable function of, then, d df dg (i) (f ± g) = ± d d d (ii) d df (cf) = c, where c is an constant d d (iii) d dg df (f g) = f + g known as PRODUCT RULE d d d (iv) d d æ è df dg g æ ö f æ ö f ö è d è d = g g where g¹ 0 known as QUOTIENT RULE (v) If = f(u) & u = g () then 5. DERIVATIVE OF STANDARDS FUNCTIONS : du =. known as CHAIN RULE d du d n n (i) D( ) = n. ; ÎR, nîr, > 0 (ii) D (e ) = e (iii) D(a ) = a. l na a > 0 (iv) D ( ln ) = (v) D(loga ) = loga e (vi) D (sin) = cos (vii) D (cos) = sin (viii) D (tan) = sec (i) D (sec) = sec. tan () D (cosec) = cosec. cot (i) D (cot) = cosec (ii) D (constant) = 0 d where D = d Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. #
6. INVERSE FUNCTIONS AND THEIR DERIVATIVES : (a) Theorem : If the inverse functions f & g are defined b = f () & = g () & if f () eists & f ' () ¹ 0 then d æ ¹ 0 d, then = / ö èd (b) Results : ' g () = '. This result can also be written as, if eists & f () d d æ ö éd ù or. = or = / d ê ¹ 0 d ú èd ë û (i) D (sin ) =, < < (ii) D (cos ) =, < < (iii) D (tan ) =, Î R (iv) D (sec ) =, > + (v) D (cosec ) =, > (vi) D (cot ) =, ÎR + Note : In general if = f(u) then du = f'(u). d d. 7. LOGARITHMIC DIFFERENTIATION : To find the derivative of : (i) A function which is the product or quotient of a number of function OR (ii) A function of the form [f ()] g () where f & g are both derivable, it will be found convenient to take the logarithm of the function first & then differentiate.this is called LOGARITHMIC DIFFERENTIATION 8. IMPLICIT DIFFERENTIATION : f(,) = 0 (i) In order to find /d, in the case of implicit functions, we differentiate each term w.r.t. regarding as a functions of & then collect terms in /d together on one side to finall find / d (ii) In answers of /d in the case of implicit functions, both & are present. 9. PARAMETRIC DIFFERENTIATION : / dq If = f () & = g( q) where q is a parameter, then =. d d / dq 0. DERIVATIVE OF A FUNCTION W.R.T. ANOTHER FUNCTION : Let = f () ; z = g () then dz = / d dz / d. DERIVATIVE OF ORDER TWO & THREE : Let a function = f () be defined on an open interval (a, b). It s derivative, if it eists on (a, b) is a certain function f ' () [or ( / d) or ] & is called the first = f'() g'() derivative of w. r. t.. If it happens that the first derivative has a derivative on (a, b) then this derivative is called second derivative of w.r.t. & is denoted b f '' () or (d /d ) or ''. Similarl, the rd order derivative of w.r.t, if it eists, is defined b denoted b f ''' () or ''' d d æ d d è d = d ö It is also. If f () g () h () F() = l () m () n (), where f, g, h. l, m, n, u, v, w are differentiable functions of u () v () w () then ' f () g() h() F '() = l () m() n() + u() v () w () ' ' f () g() h() l '() m'() n'() + u() v () w () f () l () u'() g() m() v'() h() n() w'() Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. #
SINGLE CORRECT QUESTION. If = 4 + 4 + then ù d ú û = / is : 4 (D) none. If f () = and = f( ), then at = is equal to d (D). A function f, defined for all positive real numbers, satisfies the equation f( ) = for ever >0. Then the value of f (4) = / (D) cannot be determined æ + 4ö 4. If = f & f '() = tan then = è5 + 6 d tan 4 tan é + ù. êë 5 + 6úû ( 5 + 6) æ tan + 4 ö f tan 5 tan 6 (D) None è + 5. Let h() = f() f() and g() = f() f(4), if h'() = 5 and h'() = 7, then g'() is : 9 9 7 (D) 4 6. Let f() = n, n being a non negative integer. The number of values of n for which f (p +q) = f (p) + f (q) 7. If is valid for all p,q > 0 is : 0 (D) none of these d d æ + + è + + 4 ö = a + b, then the value of 'a' and 'b' are respectivel and and and 8. If = f() is an odd differentiable function defined on (, ) such that f () =, then f ( ) is equal to 4 (D) 0 9. If = e, find d * e ( + ) e ( ) e ( + ) e 0. If f () = e. Let g () be it's inverse then g' () at = is ln. If = then is equal to d ln. ln ( ln + ). ln (D) e ( ln + ). + (D) none of these Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. #
. If f() = log (ln ), then f () at = e is equal to /e e (D) zero. If = log ( ), find d + log( ) + log( ) + log( ) 4. If = log + +, find d + ( + ) + + ( + ) + + ( + ) 5. If. =, then equals to d (ln ) (ln ) (ln ) (ln + ) (l n + ) (ln ) (D) ( ln + ) (ln + ) d 6. If =, then + 4 4 + + equals to 0 4 + 4 + 7. If + + + = 0, then d is equal to ( +) ( +) ( + ) (D) ( + ) 8. If a + h + b = 0, then d is equal to (D) 9. If sin () + cos () = 0 then d is equal to (D) Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. # 4
0. If sin + =, find d ( cos ) cos + ( cos + ) cos+ ( cos + ) cos+. If = tan (sin ), find d cos + sin cos + sin cos + sin F HG a+ bcos. If = cos b+ acos b a b+ acos I K J, find d. If = tan (sec + tan ) find d b a b+ acos a b bacos / (D) / 4. If = sin æ è + ö and = d ( ) + p, then p is equal to 0 F HG 5. If = sin + I K J, then d at = p is sin (D) p 8 + 4 4 p + 4 8 p + 4 (D) Does not eists 6. If = a cos t, = a sin t, find d tan t tan t cot t 7. Diff. sin w.r.t. sec cos cos cos 4 sec tan 4 sec tan 4 sec tan 8. If =, then the derivative of w.r.t. is + + + + (D) none of these Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. # 5
9. Derivative of æ tan è + tan ö w.r.t. tan is: æ ö è + tan ( + tan ) (D) (+ tan ) æ ö 0. Let f () be defined for all > 0 & be continuous. Let f () satisf f = f() f() for all, & f (e) =. Then: è æ ö f () is bounded f as 0.f() as 0 (D) f () = ln è. The funcion f() = e +, being differentiable and one to one to one, has a differentiable inverse f (). The value of ( f ) ln d d at the point f( l n) is d. If = at, = at, then is equal to d t at 4 t. Let f() be a polnomial in. Then the second derivative of f(e ) w.r.t. is f (e ). e + f (e ) f (e ). e + f (e ). e f (e ) e (D) f (e ). e + f (e ). e (D) none (D) at d 4. If = + e then is : e e (+ e ) e (+ e ) (D) ( + e ) 5. A curve is parametricall represented b = R( cosq) & = R(q sinq), then d (D) R 4R R 4R 6. If = lnt & = t then ''() at t = is at q = p is 4 (D) none 7. 4 + + 7 + 4 5 4 d = 0, then d at (, ) is (C* ) 7 (D) 0 d i is 8. If =sint and =sint, then the value of K for which + K = 0 d d 6 (D) 9 Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. # 6
9. If f is twice differentiable such that f ()= f(), f ()=g() h () = [f()] + [g()] and h(0) =, h()=4 then the equation = h() represents : a curve of degree a curve passing through the origin a straight line with slope (D) a straight line with intercept equal to æ ö 40. If = cosecq sinq ; = cosec n q sin n q, then ( + 4) è d n equals to n n n (D) 4n 4. If = /5 + /5, then ( ) d + equals to d d 5 5 5 (D) + 5 4. If = sin(m sin ), then ( ) is equal to m m m æ ö d 4. If = ln, then è a + b d is equals to æ ö è d æ ö è d æ ö + è d 44. If æ d ö æ d = P(), is a polnomial of degree, then ö è. equals : d è d P () + P () P (). P () P(). P () (D) a constant æ ö æ ö æ ö 45. If = tan + tan è + + + tan è + + +... to n terms,then is equal to è + 5 + 7 d + n + ( + n) + + + (n + ) + 46. If + =, then ; ' ' (' ) + = 0 ' ' + (') + = 0 ' '(') = 0 (D) ' ' + (' ) + = 0 [JEE 000, Screening, out of 5] Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. # 7
47. If ln ( + ) = 0, then (0) is equal to [JEE 004, Screening] 0 (D) 48. If = () and it follows the relation cos + cos = p then ''( 0 ) [JEE 005, Screening] p (D) p F HG I F + KJ HG F H G I K J I KJ F 49. If f''() = f() and g() = f () and F()= f H G I K J g and given that F(5) = 5, then F(0) is equal to 5 0 0 (D) 5 [JEE 006, (, ) out of 84] 50. d equals [JEE 007, (, ) out of 8] F HG d I KJ F H G I K J F H G I K J d d F HG AIEEE I d KJ F H G I d K J F (D) H G I K J F H G I K J d d 5. If = ( + + ) n, then ( + d ) d + is [AIEEE 00] d n n (D) 5. If sin = sin (a + ), then is [AIEEE 00] d sin a sin (a + ) sin (a + ) sin a sin a sin (a + ) (D) sin (a ) sin a 5. If = e, then is [AIEEE 00] d + + log log + log log not defined (D) (+ log) 54. If = e + e +..., > 0 then is [AIEEE 004] d + (D) + 55. If m n = ( + ) m + n, then is [AIEEE 006] d + (D) Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. # 8
56. Let be an implicit function of defined b cot = 0. Then equals : [AIEEE 009] log log (D) 57. Let f : (, ) R be a differentiable function with f(0) = and f (0) =. Let g() = [f(f() + )]. Then g (0). 4 0 (D) 4 [AIEEE 009] MULTIPLE CORRECT QUESTION 58. Let = + + +... then d + 4 (D) + 59. If + = + then has the value equal to : d (D) ( ) ( ) 60. If = cos t, = log e t then p = at t = d p 4 p = at t = d p 44 p = at t = d p 6 (D) d = p at t = 6 p 6. If f() = ( 4) ( 5), then f () is equal to + 9, for all Î R 9 if > 5 + 9 if 4 < < 5 (D) not defined for = 4, 5 Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. # 9
. The value of at which the first derivative of the function +. If = log (log(sin + )), find d u. If = and u =, find u + d 4. If + 8 + = 64, find d sin 5. If =, find d F HG I K J w.r.t. is /4, are a( t ) bt 6. If = = & + t + t, find d q q 7. If = ae (sinq cos q), = ae (sinq + cos q), find the value of d at q p = 4 8. If + = ( + ) +, find d 9. Diff. sin w.r.t. (sin ) 0. If = + log, find d. If = e, then find in terms of onl. d. If = F HG ab tan tan, find a b a+ b d. If = sin (sin ), and d + tan + f ( ) =0, then f () equal d d d ( + ) 4. If tan (/ ) + = e Prove that =, > 0. d ( ) I KJ n n 5. If = sec qcos q ; = sec q cos q æ ö, then show that ( + 4) n ( + 4) = 0 è d 6. If sin + = 0. Find the value of K if cos æ ö + + sin + K = 0 è d sina 7. If sin = sin (a + ), show that = = d cosa+ sin (a + ) sina 6 6 8. If + = a.( ), prove that = 6. d 6 Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. # 0
NCERT BOARD QUESTIONS. The derivative of log is. If = +, then d d. If + = a( ) then 4. If = cot + sec, then d d d d 5. If + d = t, 4 + 4 = t +, then t t d 6. If e + e + = 0, then 7. The derivation of 8. d d æ ö æ ö sin w.r.t. cos is è + è + é êtan d ë d sec tan 9. If = then sec + tan 0. If = sin sin then d d d d. If = (t + ) (t + ), then d d t æ cos öù ú è+ sin equals û. If = q q and = q + q, then d d = 4. If n m + m n = 6 then d n d m = 5. If = ep. tan æ ö, then è d d. Derivative of sin w.r.t. cos is æ d ö 4 6. cos = d è 7 Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. #
7. Derivative of sec æ ö w.r.t è + + at = is æ 8. If cos ö è + 9. d ìï æ + öüï ísin cot d ý ïî è ïþ = = log a then d = 0. If = (a )( b) (a b) tan æa ö b è, then d equals. If sin ( + ) + cos ( + ) = log ( + ), then d is. Find the value of the epression d on the ellipse + 4 =. [Ans. 9 4 ]. If 5f() + f æö è = + and = f (), then find d at =. 4. = (sin + cos), then is d 5. = log cos, then sin d is 6. = 7. = sin tan e æ e è e a a æ ö è e + e a a ö, then is d, then is d 8. If n 4 = ( + ) ( + )( + )...( + ) then at = 0 is d 9. If log( ) = + + then the value of (0) is : 0. If ( ) m = + + then ( ) + + m=. If + + = cthen d is :. If = sin (sin), and d tan f() 0 d + + = then f() is : d. If = cos t cos t and = sin t sin t then the value of att = p is : d 4. If + + =, find. d Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. #
OBJECTIVE... 4. 5. 6. 7. 8. 9. 0.... 4. 5. (D) 6. 7. 8. 9. 0.... (D) 4. (D) 5. 6. 7. 8. 9. 0. (D).. (D). (D) 4. 5. (D) 6. 7. (D) 8. (D) 9. 40. (D) 4. 4. 4. 44. 45. 46. 47. 48. 49. 50. (D) 5. n 5. 5. (D) 54. 55. (D) 56. (D) 57. 58. (A,C, D) 59. (A,B,C,D) 60. (A, D) 6. (B, C, D) SUBJECTIVE. +. cot + ( + ) log sin +. ( + ) ( + 8) 4. 8+ 5. + sin b ( t ) ( ) / 6. at 7. 8. + log ( + ) { + log( + )} [( + ){ + log( + )} log ] 9. sin [( / ) sin + cos log ] (sin ) [ cot+ logsin ] 0. log ( + log. ) ln. = (+ ln). a+ b cos. cos sin (sin ) NCERT BOARD QUESTIONS. æ ö. è. = d 4. 0 Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. #
5. 6. e + ( + ) (e + ) 7. 8. 9. sec (sec tan ) 0... 4. + cosec. m7n 7mn + ( t+ ) ( t+ ) 5. [ + tan (log )] + sec (log ) 6. 9 7. 0 8. 9. 0. æa ö è b.. 9. 4 7 8 é(cos sin ù 4. (sin + cos ) ê + log(sin + cos) ú ë sin + cos û 5. d cos = logsin + log(cos ) (logsin ) sin a 6. a a e + e 7. + 8. 9. 0 0. ( ) + + = m.. c. cos. sin (sin ). 4. ( + ) ( + ) Arride learning Online Elearning Academ A479 Indra Vihar, Kota Rajasthan 4005 Page No. # 4