CHAPTER 7 SUCCESSIVE DIFFERENTIATION

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1 CHAPTER 7 SUCCESSIVE DIFFERENTIATION TOPICS:. Successive differetiatio-th derivative of a fuctio theorems.. Fidig the th derivative of the give fuctio.. Leibitz s theorem ad its applicatios.

2 SUCCESSIVE DIFFERENTIATION Let f be a differetiable fuctio o a iterval I. The the derivative f is a fuctio of ad if f is differetiable at, the the derivative of f at is called secod derivative of f at. It is deoted by f () or f () ().similarly, if f is differetialble at, the this derivative is called the rd derivative of f ad it is deoted by f () (). Proceedig i this way the th derivative of f is the derivative of the fuctio f (-) () ad it the deoted by f () (). () d y If y f() the f () is deoted by or D y or y () or y d ( ) ( ( ad ) ) f h f f lim h 0 h If f() (a b) m, m R, a b > 0 ad N the () m f () m(m )(m )...(m )(a b) a Note : If y (a b) m the y m(m )(m ) (m )(a b) m a. COROLLARY If f() (a b) m, m Z, m > 0, N the (i) m < f () () 0, (ii) m f () ()! a (iii) m > f () m! m () (a b) a. (m )! COROLLARY If f() is a polyomial fuctio of degree less tha where N the f () () 0. If f() a b the f () () ( )!a (a b).(i.e., If y a b y ( )!a (a b) ) If f() log a b ad N the f () ( ) ( )!a () (a b). i.e., y log ab y ( ) ( )!a (a b)

3 If f() si(a b) ad N the If f() cos(a b) ad N the If f() e ab ad N the () π f () a si a b. () π f () a cos a b. () a b f () a e. If f() c ab, c > 0 ad N the () a b f () a c (logc). If f() θ ad If f() θ ad a e si(b c) ad N the r a b, θ Ta a b a e cos(b c) ad N the r a b, θ Ta a. b () a Note: If f, g are two fuctios i havig their th derivatives the (f ± g) () () () () f () ± g (). f () r e si(b c θ ) where a r cos θ, b r si () a f () r e cos(b c θ ) where a r cos θ, b r si Note: If f is a fuctio i havig th derivative ad k R the () () (kf ) () kf ().

4 EXERCISE 7 (a). Fid the th derivative of si. si si Sol: we kow that si si si si Differetiate times w.r.t, ( si ) ( si si ) d d π π.si si z. Fid the th derivative of si 5. si.? Sol: let y si5.si ( si5.si) y ( cos cos8 ) y ( cos cos8) Differetiate times w.r.t, d y ( cos cos8) d d π π y cos 8.cos 8 z d. Fid th derivative of e.cos.cos Sol: cos.cos ( cos.cos ) ( cos cos ) e Let y ( cos cos) Differetiate times w.r.t,. If d y ( e cos e cos ) d e e y ( 0 ) cos( ta ) ( ) cos( ta / π ) z 0 cos( ta ) cos y fid y ( )( ) y ( partial fractios) Sol: Give Differetiate times w.r.t, ( )! ( )! y ( ) ( ) ( )! ( ) ( )

5 5. If y, fid y A B Sol: Let A( ) B( ) -----() 5 A 5 A B B 5 y I (),Put I (), Therefore, Differetiate times w.r.t., d 5 y d ( ) ( ) 5 ( )! ( )! y ( ) ( )! 5 ( ) ( ). Fid the th derivative of (i) ( ) ( ) (iv) (v) Sol: i) Let y ( ) ( ) Resolvig ito partial fractios A B C (vi) Log ( 9) ( ) ( ) ( ) A( )( ) B( ) C( ) ( (ii) ( )( ) (iii) ( )( ) B B B I, put I, put C( ) C C Equatig the co. efficiet of A B 0 A Therefore, y ( ) ( ) ( ) Differetiate times w.r.t., d y d ( ) ( ) ( )

6 ( )! ( )( )...( ) ( )! y ( ) ( ) ( ) ( )! ( ) ( )! ( )!. ( ) ( ) ( )! ( ) ( ) ( ) (ii) y ( )( ) A B C Resolvig ito partial fractios ( )( ) ( ) A ( ) B( )( ) C( ) ---() I () put A( ) 9A A 9 (iii) As: I () put C( ) C C Equatig the co efficiet of I () A B 0 B A 9 y 9( ) 9( ) ( ) Differetiate times w.r.t., d y d 9( ) 9( ) ( ) ( )! ( )! ( ) ( ) y ( )! 9( ) 9( ) ( ) 9 ( ) 9 ( ) ( ) y ( )( ) ( )! ( ) ( ) ( iv) As: ( )! y cos ( ) si( ) r θ θ (v) y! As: ( ) ( )

7 (vi) y log( 9) Give y log ( 9) log[ ][ ] log( ) log( ) Differetiatig times, d y ( log( ) log( ) ) y d!!! ( ) ( ) the show that yy y c d a b y c d. If y a b Sol: Give Differetiate w.r.t., dy ( ) ( ) d ( c d) c d b a b.d bc bd ad bd bc ad y ( c d) ( c d ) Agai diff. w.t.t, ( bc ad)( ) d d ( bc ad) y ( c d) ( c d) Diff.wrt., we get d ( bc ad)( ).d 6d ( bc ad) y ( c d) ( c d) ( bc ad) 6d ( bc ad) L.H.S. yy. ( c d) ( c d) ( ad) ( c d) d bc y R.H.S. ( ) 6 ( c d) d bc ad. If y si (si ), the show that Sol: Give y si (si) Diff. wrt, y cos( si) cos Diff. wrt, y cos si ( si ) cos cos( si ) si cos.si ( si ) si.cos( si ) LHS y ( ta) y ycos y ta y ycos 0 cos si ( si ) si.cos( si ) cos ( si ) si.cos ( si ) 0 RHS. si cos

8 If y a b, the show that y ( y ). Sol: y a b Diff. wrt. X, ( ) y a b Diff. wrt, ( ) ( ).y. a. b. ( a b ) ( ) y a. b. ( ) y 5. If y ae be, the show that y Sol: y ae be y y a..e b.e y a e b.e ( ae be ) y y k 6. If y e ( a cos bsi ) the show that k Sol. y e ( a cos bsi ) Differetiatig w.r.to. k k y ky y 0 y e [ asi bcos] [ ] k k y e a si bcos y k l k e acos bsi y y.e ( asi bcos) k Differetiatig w.r.to. k k k y y.e k k y y y ( a si bcos ).e [ a.co b si ] k k y y y k.

9 k y ky y 0 7. If f() (-a) φ where φ is a polyomial with ratioal co-efficiet, the show that f(a) 0 f ( a ) ad f ( a) φ ( a) Sol: Give f ( a) φ Diff. wrt. X, ( ) φ ( ) φ f' a ' a Diff.wrt., φ φ φ φ ( a) φ ( a) φ φ f a a a Now f a a a φ a 0. φ a 0 Ad f' a 0. φ a 0. φ a f a 0 f a f a 0. φ a.0φ a φ a φ a. 8. If y e.cos, the show that y y 0 Sol: y e.cos Here a, b ad y a e cos(b c) a a b e cos(b c θ ) where y θ Ta a. b y e cos( ta ) Now y e cos( ta ) π y e cos( ) y e cos y y y y 0

10 9. If y ta, the show that y cos y 0. If y log( ) Sol: y log, the show that Differetiatig w.r.to, y.y y y y y... If y a cos (b ) si, the show that y y cos Sol: y a cos ( b ) si () Differetiatig w.r.to y asi b.cos si Agai differetiatig w.r.to ( ) si cos y acos cos b ( ) cos acos b.si cos y ( from ) Hece y y cos. If y a be, the show that y y 0. If y a b log, the show that Sol: Give y a blog Differetiatig wrt. X, b y a, b Diff. wrt., y log y y y 0

11 Now L.H.S ( log ) y y y b b ( log ) a b log a b a blog - b log b b log 0 R.H.S a blog. If y a cosec (b ), the show that yy y y Sol: y a cosec (b ) -----() Differetiatig w.r.to y acos ec b cot b y y.cot(b ) ----() Diff.with respect, y ycos ec b cot b. y y cos ec ( b ) y cot ( b ) y cos y y y ec ( b ) cos yy y ec b y cos cot yy y ec b y b y 5. If 5 ay b, the show that 5yy y. Sol: Give 5 ay b y b a 5 y b ( fidig the th root) a 5 5 y.. b a Differetiatig w.r.to, /

12 5 y.. b a Diff.wrt., / Now 5 a 6a 5 L.H.S 5yy 5 ( b) ( b) 6 5 b ( ) a 5 a b y 6. If y 6() (AB)e, the show that y 6y 9y 5 8 III.. If θ [ π, π ] be such that cos θ si θ R, ad the prove that i) th derivative of ta is ( ) ( )! si θ ( ) / ii) th derivative of ta is ( )!si θsiθ iii) th derivative of ta is ( ).( )!si θ( siθ ) Sol: i) let y ta, put ta θ the θta- y ta taθ ta θ ta ta θ θ ta Therefore, y ta Differetiate wrt. X, y y i i i ( partial fractios)

13 Differetiatig - times w.rt., y D i i i ( ) ( ) ( ) ( )!! () i i i let rcos θ ad rsi θ si θ r The i r ( cos θ isi θ ) Now r r ad ta θ θ ta Now ( i) r ( cos θ isi θ ) r θ i θ ( i) ( cos si ) r i i ( i) r ( cos θ isi θ ) ( cos θ si θ) r i θ ( i) ( i) i ( si ) () { } From () ad () y ( ) ( )! r ( isiθ) ( )! siθ r ( ) ( )!si θ.si ( θ ) si r θ ( ) ( )!si θ si θ ( ) ( )!, θ ta

14 ii) Let y ta ta () ta proceed as above problem iii) Let y ta ta as above problem.. If a hy by, the show that d y h ab d h by ( ) Sol: Give equatio a hy by () Differetiatig w.r.to, dy dy a h. y. b.y 0 d d dy h by a hy d ( ) ( ) ( a hy) ( ) dy d h by Differetiatig w.r.to () dy dy ( h by) a h ( a hy) h b. d d d y d h by ( ) ( h by) h a hy b a hy ( h by) a ( a hy) h h by h by ( h by)( ah aby ah h y) ( a hy)( h bhy ah bhy) ( h by) ( )( ) ( )( ) h by ab h y a hy h ah ( h by) ( )( h ab hy by a hy ) ( h by) h by h by h by h ab a hy by h ab. h ab

15 b. If y ae cos( c d) y by b c y 0 the show that b Sol: y ae cos( c d) () Diff. wr.t, b b y ae. si c d c acos c d e b ( b). y ace. b.si( c d) b y by ace. si( c d) () Differetiatig w.r.to ( b b ) y by ac e cos c d. c b e.si c d ). b c ae cos( c d) b ac. e b si ( c d ). b b c ae cos ( c d) b aee. si( c d) c y b( y by) from () ad () c y by b y y by b c y 0. d, prove that d. if y ( log ) y! log... > 0 Sol. d y (.log ) d ( ) (.log ) D D D ( log ) D D D log!. y log y y! dividig with!,

16 y y ()!! ( ) I (), put,,,,5-----, the y y! 0! 0 y y!! y y (! ) ( ) ( ) y y!! ( ) Addig above equatios, we get y y0...!! y y0...!! y y0...!! y0 y!...! 0 0 y! log..., here y0 D log log

17 LEIBNITZ If f, g are two fuctios i havig th derivatives the () () ( ) 0 ( ) () ( r) (r) r (fg) () C f ()g() C f ()g () C f ()g ()... C f ()g () ()... Cf ()g (). Proof : Let S() be the statemet that 0 (fg) () C f ()g() C f ()g () ( r) r r... C f ()g ()... C f()g () Now (fg) () f()g () g()f () ( product rule) From above statemet, (fg) C f ()g() C f()g () S() is true. Assume that S(k) is true. 0 (k) k (k) k (k ) k (k r) (r) k (k) 0 r k (fg) () C f ()g() C f ()g ()... C f ()g ()... Cf()g () Now (k ) (k) (fg) () [(fg) ()] d k (k) k (k ) k (k r) (r) k (k) ( C0f ()g() Cf ()g ()... Crf ()g ()... Ckf()g ()) d k (k ) k (k) k (k) k (k ) () C f ()g() C f ()g () C f ()g () C f ()g () k (k r ) (r) k (k r) (r ) k (k) k (k ) Crf ( r k k )g () C f ()g ()... C f ()g () C f()g () (k ) k k (k) k k (k ) () k k f ()g() C0 C f ()g () C C f ()g ()... Cr C r (k r ) (r) (k ) f ()g ()... f ()g () (k ) (k ) (k ) (k ) (k ) (k ) () (k ) (k r) 0 r C f ()g() C f ()g () C f ()g ()... C f () (r) (k ) (k ) k g ()... C f ()g () S(k ) is true. By priciple of Mathematical Iductio S() is true for all N. () () ( ) ( r) () () 0 r (fg) () C f ()g() C f ()g ()... C f ()g ()... Cf()g ()

LIMITS AND DERIVATIVES

Capter LIMITS AND DERIVATIVES. Overview.. Limits of a fuctio Let f be a fuctio defied i a domai wic we take to be a iterval, say, I. We sall study te cocept of it of f at a poit a i I. We say f ( ) is