Differentiate the following w.r.t. x:`(xcosx)^x+(xsinx)^(1/x)`

1.	Differentiate the following w.r.t. x:`(xcosx)^x+(xsinx)^(1/x)`
Answer» `"Let" y = (x "cos" x)^(x) + (x"sin"x)^(1//x)` `"Let" u = (x "cos" x)^(x) " and " v = (x"sin"x)^(1//x)` `therefore y=u+v` `rArr (dy)/(dx) = (du)/(dx) + (dv)/(dx) " "....(1)` `"Now "u = (x"cos" x)^(x)` `rArr "log" u = "log"(x"cos"x)^(x) = x("log"x + "log cos"x)` `rArr (1)/(u)(du)/(dx) = x * (d)/(dx)("log" x + "log cos"x) + ("log"x + "log cos"x)(d)/(dx)x` `rArr (du)/(dx) = u[x((1)/(x)-("sin"x)/("cos"x)) + ("log"x + "log cos"x) * 1]` `rArr (du)/(dx) = (x"cos"x)^(x)[1-x"tan"x + "log"x + "log cos"x]` `"and "v = (x"sin"x)^(1//x)` `rArr "log" v = "log"(x"sin"x)^(1//x) = (1)/(x)("log"x + "log sin"x)` `rArr (1)/(v)(dv)/(dx) = (1)/(x)(d)/(dx)("log"x + "log sin"x) + ("log"x + "log sin"x)(d)/(dx)((1)/(x))` `rArr (dv)/(dx) =v[(1)/(x)((1)/(x) + ("cos"x)/("sin"x)) - (1)/(x^(2))("log"x + "log sin"x)]` `rArr (dv)/(dx) = (x"sin"x)^(1//x)[(1 + x"cot"x - "log"(x"sin"x))/(x^(2))]` From equation (1) `(dy)/(dx) = (x"cos"x)^(x)[1 - x"tan"x + "log"x + "log cos"x] + (x"sin"x)^(1//x)[1 - x"tan"x + "log"(x"sin"x)]`

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