What are the long steps to determine the derivative of x plus the square root of x?

Question

d/dx x+x^1/2

Answer 1

d/dx (x + √x)

This is your question?

Realize that √x is the same as x^(½)

And that d/dx [ f(x) + g(x) ] = d/dx f(x) + d/dx g(x)

And that d/dx x^b = b·x^(b-1)


So,
d/dx (x + √x) =
d/dx (x) + d/dx (x^(½)) =
1·x^0 + ½·x^(-½) =
1 + ½·x^(-½) =
1 + ½·√(1/x)

Answer 2

f (x) = x + x ^(1/2)
f `(x) = 1 + (1/2) x ^(-1/2)
f `(x) = 1 + 1 / ( 2x^(1/2) )