Using the definition of the derivative, find the derivative of f(x) = x + sqrt(x)

Using the definition of the derivative, find the derivative of f(x) = x + sqrt(x).

Is it 1 + [1 / 2sqrt(x)]?

Almost. You need to rationalize the fraction.
http://www.flickr.com/photos/dwread/7473…

You are technically correct, but you can simplify further by rationalizing the denominator:
1 / 2(√x)
1*√x / 2(√x)(√x)
√x / 2x

The derivative of f(x) = x+√x is 1 + (√x / 2x).

1 + 1/(2 Sqrt[x])

Yes

That is correct.