1.

Differentiate w.r.t. x the function `x^x+x^a+a^x+a^a ,`for some fixed `a > 0 a n d x > 0`.

Answer» Let `y=x^(x)+x^(a)+a^(x)+a^(a)`
`implies(dy)/(dx)=(d)/(dx)x^(x)+(d)/(dx)x^(a)+(d)/(dx)a^(x)+(d)/(dx)a^(a)`
`implies(dy)/(dx)=(d)/(dx)x^(x)+a*x^(a-1)+a^(x)loga+0 " " `...(1)
Let `u=x^(x)`
`implieslogu=logx^(x)=xlogx`
`implies(1)/(u)(du)/(dx)=x*(d)/(dx)logx+logx*(d)/(dx)x`
`implies(du)/(dx)=u(x*(1)/(x)+logx*1)=x^(x)(1+logx)`
From equation (1)
`(dy)/(dx)=x^(x)(1+logx)+a*x^(a-1)+a^(x)loga`


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