just use the difference quotient, and recall how to "rationalize" expressions involving roots:
√x - a = (√x-a)(√x+a)/(√x+a)
Find the derivative of f'(x) of f(x)=4sqrt(x) using limits
3 answers
I did use the difference quotient but got a really weird answer. This was on a test I took earlier this morning and I know we get test corrections once the professor passes the test back
√(x+h)-√x
------------
h
(√(x+h)-√x)(√(x+h)+√x)
--------------------------
h(√(x+h)+√x)
(x+h)-x
-------------------
h(√(x+h)+√x)
h
---------------------
h(√(x+h)+√x)
The h's cancel, and you have
1/(√(x+h)+√x)
As h->0, that is 1/(√x+√x) = 1 / 2√x
------------
h
(√(x+h)-√x)(√(x+h)+√x)
--------------------------
h(√(x+h)+√x)
(x+h)-x
-------------------
h(√(x+h)+√x)
h
---------------------
h(√(x+h)+√x)
The h's cancel, and you have
1/(√(x+h)+√x)
As h->0, that is 1/(√x+√x) = 1 / 2√x