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        Differentiate it please as earlier as possible and explain it briefly
10 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 .

9 months ago
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