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Differentiate it please as earlier as possible and explain it briefly

Alka Sinha , 6 Years ago
Grade 12
anser 1 Answers
Samyak Jain

Last Activity: 6 Years ago

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