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"Find "(dy)/(dx)" for "y=log(x+sqrt(a^(2...

`"Find "(dy)/(dx)" for "y=log(x+sqrt(a^(2)+x^(2))).`

Text Solution

Verified by Experts

`y=log (x+sqrt(a^(2)+x^(2)))`
`"Then "(dy)/(dx)=(d)/(x){log (x+sqrt(a^(2)+x^(2)))}`
`=(1)/(x+sqrt(a^(2)+x^(2)))(d)/(dx)(x+sqrt(a^(2)+x^(2)))`
`=(1)/(x+sqrt(a^(2)+x^(2)))xx{1+(1)/(2)(a^(2)+x^(2))^(-1//2)(d)/(dx)(a^(2)+x^(2))}`
`=(1)/(x+sqrt(a^(2)+x^(2))){1+(1)/(2sqrt(a^(2)+x^(2)))xx2x}`
`(1)/(x+sqrt(a^(2)+x^(2)))xx(sqrt(a^(2)+x^(2))+x)/(sqrt(a^(2)+x^(2)))`
`=(1)/(sqrt(a^(2)+x^(2)))`
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