Question

, for some fixed and

Answer 1

Let, y= x x + x a + a x + a a .

Let, u= x x ,v= x a ,w= a x .

Substitute the above value in given equation.

y=u+v+w+ a a (1)

Differentiate both sides with respect to x.

dy dx = d dx ( u+v+w+ a a ) = du dx + dv dx + dw dx + d dx ( a a ) = du dx + dv dx + dw dx +0 dy dx = du dx + dv dx + dw dx (2)

Let, u= x x .

u= x x logu=log x x logu=x×logx

Differentiate both sides with respect to x.

d dx ( logu )= d dx ( xlogx ) 1 u du dx = d dx ( x )×logx+ d dx ( logx )×x =logx+ 1 x ×x 1 u du dx =logx+1

Further simplify.

du dx = x x ( 1+logx )

Let, v= x a . Differentiate both sides with respect to x.

v= x a dv dx =a x a−1

Let, w= a x . Differentiate both sides with respect to x.

w= a x dw dx = a x loga

Substitute the above values du dx , dv dx , dw dx in equation (2).

dy dx = x x ( 1+logx )+a x a−1 + a x loga

Thus, the above equation is differentiation of y= x x + x a + a x + a a .