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Find the values of (dy)/(dx), if y = x^(tanx)+sqrt((x^(2)+1)/(2)). |
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Answer» Taking ` u=x^(tanx)` and `v = sqrt((x^(2)+1)/(2))` `logu = tanxlogx"...."(ii)` and `v^(2) = (x^(2)+1)/(2)"....."(III)` On differentiating Eq. (ii) w.r.t.x, we get `1/u.(du)/(dx)= tanx.(1)/(x)+logx.sec^(2)x` `rArr (du)/(dx)=u[(tanx)/(x)+logx.sec^(2)x]` `=x^(tanx)[(tanx)/(x)+logx.sec^(2)x]"....."(iv)` ALSO, differentiatingEq. ( ii) w.r.t.x, we get `2v.(dv)/(dx)=1/2(2x)rArr (dv)/(dx)= (1)/(4V).(2x)` `rArr (dv)/(dx) = (1)/(4.sqrt((x^(2)+1)/(2))).2x= (x.sqrt(2))/(2sqrt(x^(2)+1))` `rArr (dv)/(dx)=(x)/(sqrt(2(x^(2)+1)))"...."(v)` Now, `y=u+v` `:. (dy)/(dx)=(du)/(dx)+(dv)/(dx)` `=x^(tanx)[(tanx)/(x)+logx.sec^(2)x]+(x)/(sqrt(2(x^(2)+1)))` |
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