....... y = [ x + √(x²-1) ]ⁿ
∴ ln y = n· ln [ x + √(x² - 1) ]
∴ d/dx( ln y ) = n· d/dx( ln u ), ...... where : u = x + √(x²-1)
∴ (1/y).(dy/dx) = n·(1/u).(du/dx)
∴ dy/dx = (ny/u). d/dx[ x + √(x²-1) ]
. . . . . . . = (ny/u). { 1 + [ 1/(2√(x²-1)) ].(2x) }
. . . . . . . = (ny/u). { 1 + [ x / √(x²-1) ] }
. . . . . . . = (ny/u). { [ √(x²-1) + x ] / √(x²-1) }
. . . . . . . = (ny/u). [ u / √(x²-1) ]
. . . . . . . = (ny) / √(x²-1) .......................................... Ans.
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....... y = [ x + √(x²-1) ]ⁿ
∴ ln y = n· ln [ x + √(x² - 1) ]
∴ d/dx( ln y ) = n· d/dx( ln u ), ...... where : u = x + √(x²-1)
∴ (1/y).(dy/dx) = n·(1/u).(du/dx)
∴ dy/dx = (ny/u). d/dx[ x + √(x²-1) ]
. . . . . . . = (ny/u). { 1 + [ 1/(2√(x²-1)) ].(2x) }
. . . . . . . = (ny/u). { 1 + [ x / √(x²-1) ] }
. . . . . . . = (ny/u). { [ √(x²-1) + x ] / √(x²-1) }
. . . . . . . = (ny/u). [ u / √(x²-1) ]
. . . . . . . = (ny) / √(x²-1) .......................................... Ans.
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