Funksiyaning hosilasi va differensiali

Mashqlar

10. 
    Ushbu
    

 0, agar x ratsional son bo'lsa, ^{f (x)  } ², agar x irratsional son bo'lsa x
funksiyaning x  0 nuqtadagi hosilasi topilsin.

10. 
    1) Agar f (x) funksiya x₀ nuqtada f (x₀ ) hosilaga ega bo’lsa, u holda shu nuqtada funksiyaning o’ng hosilasi f (x₀  0) va chap hosilasi f (x₀  0) mavjut hamda f (x₀  0)  f (x₀ 0)  f (x₀ ) bo’lishi isbotlansin. 2) Agar f (x) funksiya x₀ nuqtada o’ng hosila f (x₀  0) chap hosila f (x₀  0) larga ega bo’lib, f (x₀  0)  f (x₀  0) bo’lsa, u holda shu nuqtada funksiyaning f (x₀ ) hosilasi mavjud va f (x₀ )  f (x₀  0)  f (x₀  0) bo’lishi isbotlansin.
    
11. 
    Bir tomonli hamda cheksiz hosilalarning geometrik ma’nosi keltirilsin.
    
12. 
    Ushbu 
    

 0, agar x  0 bo'lsa,

f (x)  ^_ ¹ , agar ¹  x  ¹ bo'lsa,
_ n n 1 n
 1 1 1
_ n , agar  n  x   n _1 bo'lsa
funksiyaning x  0 nuqtada differensiallanuvchi bo’lishi isbotlansin.
6.14. Agar f (x) va  (x) funksiyalar xa nuqtada hosilaga ega bo’lmasa,
f (x)
f (x)  (x), f (x) (x), ( (x)  0)  (x)
funksiyalar shu nuqtada hosilaga ega bo’ladimi? Misollar keltirilsin.

15. 
    Ushbu
    

dy
 f (x)
dx
belgilashda chap tomondagi ifodani kasr deb qarash mumkinmi?

15. 
    Agar f (x) va g(x) funksiyalar x(a,b) nuqtada ntartibli
    

f ^n(x),g^n(x) hosilaga ega bo’lsa, f (x) g(x)
funksiyaning n tartibli hosilasini topish formulasi keltirib chiqarilsin.
(Odatda bu formula Leybnis formulasi deyiladi).
6.17. Ferma, Roll, Lagranj teoremalari qanday geometrik ma’noga ega?
6.18. Ushbu
1 x ^5
a). f (x)  , b). f (x)  ln
1 x x 4
funksiyalarning Makloren formulalari yozilsin.
6.19. Asimptotik formulalardan foydalanib, ushbu

lim(cos x  x2 ) x(sin1xx)
x0 2
limit topilsin.
6.20. Ushbu

f (x)  x ³
funksiya x  0 da uchinchi tartibli hosilaga egami? x ⁿ chi?