Using the definition of derivative we have:
#d/dx( root(3)x ) = lim_(h->0) (root(3)(x+h)-root(3)(x))/h#
Now use the identity:
#(a^3-b^3) = (a-b)(a^2+ab+b^2)#
with #a= root(3)(x+h)# and #b=root(3)(x)#:
#( ( root(3)(x+h))^3 - (root(3)(x))^3) = (root(3)(x+h)-root(3)(x) )( root(3)((x+h)^2)+ root(3)(x(x+h)) + root(3)(x^2))#
#( x+h) -x = (root(3)(x+h)-root(3)(x) )( root(3)((x+h)^2)+ root(3)(x(x+h)) + root(3)(x^2))#
#h= (root(3)(x+h)-root(3)(x) )( root(3)((x+h)^2)+ root(3)(x(x+h)) + root(3)(x^2))#
So:
#(root(3)(x+h)-root(3)(x) )/h = 1/( root(3)((x+h)^2)+ root(3)(x(x+h)) + root(3)(x^2))#
and then:
#lim_(h->0) (root(3)(x+h)-root(3)(x))/h = lim_(h->0) 1/( root(3)((x+h)^2)+ root(3)(x(x+h)) + root(3)(x^2)) = 1/(3root(3)(x^2))#