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If`y=sqrt(log{sin((x^2)/3-1)}),t h e nfin d(dy)/(dx)dot` |
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Answer» `y=sqrt(log{sin((x^(3))/(3)-1)}}` Putting `(x^(3))/(3)-1=v,` we get sin `((x^(2))/(3)-1)` = sin v = u. Putting log `{sin((x^(2))/(3)-1)} = log y = z, we get y =sqrt(z),z = log u.` `u=sin v, and v=(x^(2))/(3)-1.` Therefore, `(dy)/(dx)(1)(2sqrt(z)), (dz)/(du) =(1)/(u), (du)/(dv) = cos v, and (dv)/(dx)=(2x)/(3)` `Now,(dy)/(dx)=(dy)/(dz)xx(dz)/(du)xx(du)/(dv)xx(dv)/(dx)` `=((1)/(2sqrt(z))((1)/(u))(cos v) ((2x)/(3))` `=(x)/(3).(cos v)/(usqrt(log u))` `=(x cot((x^(2))/(3)1))/(3sqrt(log{sin((x^(2))/(3)-1)}})` |
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