The difference quotient is used to define the derivative as a limit.
Essentially, you find the slope of the secant line through a function at two points: (x,f(x)), and (x+h, f(x+h)). As h->0, we approach the slope of the line tangent to the function at only the point (x, f(x))
The formula for difference quotient is:
lim_(h->0) = DQ = (f(x+h)-f(x))/(h)
For this function, using a for our initial point...
(1/sqrt(a+h+2)-1/sqrt(a+2))/h = DQ
To give these factors in the numerator a common denominator, we will multiply the first factor, 1/sqrt(a+h+2), by sqrt(a+2)/sqrt(a+2), and the second factor 1/sqrt(a+2) by sqrt(a+h+2)/sqrt(a+h+2)
DQ = ((1/sqrt(a+h+2))(sqrt(a+2)/sqrt(a+2)) - 1/sqrt(a+2) (sqrt(a+h+2)/sqrt(a+h+2)))/h
Giving us...
DQ= (sqrt(a+2)/(sqrt(a+h+2)sqrt(a+2))-sqrt(a+h+2)/(sqrt(a+h+2)sqrt(a+2)))/h
We can now put this common denominator for these factors into the denominator proper:
= (sqrt(a+2)-sqrt(a+h+2))/(h(sqrt(a+2))(sqrt(a+h+2)))
Now to simplify the radicals, we multiply the numerator and denominator by the conjugate of the numerator. The conjugate of an expression sqrt(m)-sqrt(n) is simply sqrt(m)+sqrt(n). Much like how the conjugate for a-b is a+b, multiplying the numerator by the conjugate will allow us to reach the equivalent of a^2-b^2, in our case m-n
Thus:
(sqrt(a+2)+sqrt(a+h+2))/(sqrt(a+2)+sqrt(a+h+2)) * DQ
= ((a+2)-(a+h+2))/(h(sqrt(a+h+2))(sqrt(a+2))(sqrt(a+h+2) + sqrt(a+2)
Simplifying the numerator...
= -h/(h(sqrt(a+h+2))(sqrt(a+2))(sqrt(a+h+2) + sqrt(a+2)
The h factors cancel...
=-1/((sqrt(a+h+2))(sqrt(a+2))(sqrt(a+h+2) + sqrt(a+2)
If we are doing this as h->0..
lim_(x->0)DQ = -1/((sqrt(a+2))(sqrt(a+2))(2sqrt(a+2))) = -1/((a+2)2sqrt(a+2)) = -1/(2(a+2)^(3/2)
Thus f'(a) = -1/(2(a+2)^(3/2))
