So we know that
sqrt(x+2)*sqrt(x+2) = x + 2
Let's say that the first root is f(x) and the second is g(x), so we have
f(x)*g(x) = x +2
Derivating both sides we have
lim_(h rarr 0)(f(x+h)g(x+h) -f(x)g(x))/h = 1
Now, since f(x+h)g(x) - f(x+h)g(x) = 0 we can put that in a sum without changing anything, so
lim_(h rarr 0)(f(x+h)g(x+h) -f(x+h)g(x) +f(x+h)g(x) -f(x)g(x))/h = 1
Put g(x) and f(x+h) in evidence
lim_(h rarr 0)f(x+h)*(g(x+h) -g(x))/h +lim_(h rarr 0)g(x)*(f(x+h) -f(x))/h = 1
Evaluate the limit of the factors we just put in evidence
f(x)lim_(h rarr 0)(g(x+h) -g(x))/h +g(x)lim_(h rarr 0)(f(x+h) -f(x))/h = 1
The remaining limits are the definitions of f^'(x) and g^'(x), so we can rewrite it to be
f(x)g^'(x) + g(x)f^'(x) = 1
(This is actually called the product rule and is widely used for more complex functions)
But since f(x) = g(x) and f^'(x) = g^'(x) so we can further rewrite to be
2f(x)f^'(x) = 1
Isolate f^'(x) and since f(x) = sqrt(x+2) put that back in.
f^'(x) = 1/(2sqrt(x+2))
