1.

`x^("sin"x) + ("sin" x)^("cos"x)`

Answer» `"Let" y = x^("sin"x) + ("sin" x)^("cos"x)`
`"Let" u = x^("sin"x) "and " v=("sin" x)^("cos"x)`
`therefore y=u+v`
` rArr (dy)/(dx) = (du)/(dx) + (dv)/(dx) " ".....(1)`
`"Now", u=x^("sin"x)`
`rArr "log"u = "log"(x^("sin"x)) = "sin"x * "log"x`
`rArr (1)/(u) (du)/(dx) = "sin"x * (d)/(dx)"log"x + "log"x * (d)/(dx)"sin"x`
`rArr (du)/(dx) = u[("sin"x)/(x) + "log"x * "cos"x]`
`rArr (du)/(dx) = x^("sin"x)[("sin"x)/(x) + "log"x * "cos"x]`
`"and "v=("sin"x)^("cos"x)`
` rArr " log"v = "log" ("sin"x)^("cos"x)`
`= "cos" x * "log"("sin"x)`
`rArr (1)/(v) (dv)/(dx) = "cos"x (d)/(dx)"log"("sin"x) + "log"("sin"x)(d)/(dx)("cos"x)`
`rArr (dv)/(dx) = v["cos"x * ("cos"x)/("sin"x) + "log" ("sin"x) * (-"sin"x)]`
` rArr (dv)/(dx) = ("sin"x)^("cos"x)["cos"* "cot"x + "sin"x "log"("sin"x)]`
From equation (1)
`(dy)/(dx) = x^("sin"x)[("sin"x)/(x) + "log"x * "cos"x] + ("sin"x)^("cos"x)["cos"x * "cot"x - "sin"x * "log"("sin"x)]`


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