यदि `y^(2)=P(x)` घात `3` का एक बहुपद है तब `2(d)/(dx)(y^(3)(d^(2)y)/(dx^(2)))` का मान क्या है ?

Options

A. `P'`
B. `*P''`
C. `P(x)*P''(x)`
D. `*P(x)*P'''(x)`

Answer

D

Explanation

दिया है, `y^(2)=P(x)rArr2y(dy)/(dx)=P'(x)rArr(dy)/(dx)=(P'(x))/(2y)` . . .(i)
पुनः अवकलन करने पर, `2{y(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)}=P''(x)`
`rArr" "2{y(d^(2)y)/(dx^(2))+({P'(x)}^(2))/(4y^(2))}=P''(x)` [समी (i) से]
`rArr2y^(3)(d^(2)y)/(dx^(2))+({P'(x)}^(2))/(2)=P''(x)*y^(2)rArr2y^(3)(d^(2)y)/(dx^(2))=P''(x)*y^(2)-({P'(x)}^(2))/(2)`
x के सापेक्ष अवकलन करने पर,
`2(d)/(dx)(y^(3)(d^(2)y)/(dx^(2)))=P'''(x)*y^(2)+2y*(dy)/(dx)*P''(x)-(2P'(x)*P''(x))/(2)`
`=y^(2)P'''(x)+2yP''(x)(P'(x))/(2y)-(2P'(x)*P''(x))/(2)` [समी (i) से]
`=P(x)*P'''(x)+P'(x)*P''(x)-P'(x)*P''(x)=P(x)*P'''(x)`