InterviewSolution
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InterviewSolution
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If `y=(sinx)^(x)+sin^(-1)sqrtx,` find `(dy)/(dx).` |
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Answer» Let `(sinx)^(x)=u and sin^(-1)sqrtx=v`. Then, `y=u+v rArr (dy)/(dx)=(du)/(dx)+(dv)/(dx)." …(i)"` Now, `u=(sinx)^(x) rArr logu = xlog (sinx)` `rArr (1)/(u).(du)/(dx)=x.(1)/(sinx).cosx+log(sinx).1` `rArr (du)/(dx)=u{x cot x+log(sinx)}` `rArr (du)/(dx)=(sinx)^(x).{x cot x+log(sinx)}." ....(ii)"` `v=sin^(-1)sqrtxrArr(dv)/(dx)=(1)/(sqrt((1-x))).(1)/(2)x^(-1//2)=(1)/(2sqrtxsqrt((1-x)))=(1)/(2sqrt(x-x^(2)))." ...(iii)"` Using (ii) and (iii) in (i), we get `(dy)/(dx)=(sinx)^(x){x cot x+log(sinx)}+(1)/(2sqrt(x-x^(2))).` |
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