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        Differentiate it please as earlier as possible and explain it briefly
3 months ago

							You can solve this problem directly by differentiation rules but there’s an alternative simpler method of substitution!Substitute x = sin in the given expression. Then  = sin– 1x. As |x| /2  /2.dx = cos d    d / dx = 1/cos = sec          ...(1)Given expression becomes sin. /   +  log   =    sin / |cos|  +  log|cos|         =   sin / cos  +  logcos     [ for – /2  /2,  cos > 0]         =  tan + logcosd(given expression) / dx  =  {d(given expression) / d} {d / dx}                   = {d( tan + logcos)/d} {sec}             [From (1)]                   = [ sec2 + tan .1 + {1/cos}{–sin}] {sec}     [Apply uv rule of differentiation and formulae]                   = [ sec2 + tan – tan] sec                   =   sec3                ...(2)As x = sin,  x2 = sin2 = 1 – cos2. cos2 = 1 – x2   i.e.   sec2 = 1/cos2 = 1/(1 – x2), sec = 1/Substitute the value of sec in (1).So, required answer is sin– 1x . (1/)3  =  sin– 1x . 1/(1 – x2)3/2                               = sin– 1x / (1 – x2)3/2 .

2 months ago
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