find the derivative of the function using def. of derivative. state domain of function and domain of derivativ

Question

f(x)= x + square root of x

Answer 1

f(x) = x + √x

Step 1: Find the derivative.
if f(x) = a + b, then f'(x) = a' + b'.
f'(x) = 1 + 1/2(x^-1/2) = 1 + 1/(2√x)

Step 2: Find the domain of the function.
For x, the domain would be all real numbers, however, for √x, the domain is all nonnegative numbers (because the square root of a negative number is imaginary). Therefore, the function's domain is the most restrictive domain - all non negative numbers.

Step 3: Find the domain of the derivative.
The domain of 1/2√x is all positive numbers. Due to the square root, it can't allow negative numbers, and if x = 0, then you have an infinite value.