dy/dx = (a^x)(lnx)f''''(a^x), .........(1)
but f(sinx) = lnx implies f(x) = ln(arcsinx)
hence f''''(x) = (1/arcsinx) (1/ ( ( 1-x^2 ) ^ ( !/2 ) ) implies f''''(a^x) = (1/arcsin(a^x)) (1/ ((1-a ^ (2x)) ^ (1/2))) ............(2)
hence from ...(1) &.....(2) the solution is obtained but it should br noted that the given solution exist only when x belongs to (0,1].