Question

Use the definition of the derivative (don't be tempted to take shortcuts!) to find the derivative of the function

f(x) = 3 x + 2 \sqrt{x}.
Then state the domain of the function and the domain of the derivative.
f'(x) =

Domain of f(x) =

Domain of f'(x) =

Answer 1

definition of derivative: limit as h --> 0 of [f(x+h) - f(x)]/h
So it is easiest to first do f(x+h) separately:
3(x+h) + 2/sqrt(x+h)
then put all the pieces together:
f'(x) = limit as h --> 0 of [3(x+h) + 2/sqrt(x+h) - 3x - 2/sqrt(x)]/h
...

Answer 2

Notice the negative got distributed

Answer 3

Then distribute the 3 over (x+h):
f'(x) = limit as h --> 0 of [3x + 3h + 2/sqrt(x+h) - 3x - 2/sqrt(x)]/h
subtract the 3x and the -3x:
f'(x) = limit as h --> 0 of [3h + 2/sqrt(x+h) - 2/sqrt(x)]/h
Then it gets interesting...do you want me to keep going?