Question

difference quotient of sqrt(x+h)

Answer 1

difference quotient of sqrt(x+4)

Answer 2

\[f(x) = \sqrt{x+4}\] then (fx+) \[= \sqrt{x+h+4}\] subtract f(x+h) - f(x) get \[= \sqrt{x+h+4} - \sqrt{x+4}\] divide by h on both sides we get \[(f(x+h)-f(x))/h = (\sqrt{x+h+4}- \sqrt{x+4})/h\] multiply the numerator by its conjugate to get \[= ((x+h+4) - (x+4))/h(\sqrt{x+h+4}+\sqrt{x+4})\] the numerator cancels out everything but an h to get \[= h/h(\sqrt{x+h+4}+\sqrt{x+4})\] the h's cancel out to get \[= 1/(\sqrt{x+h+4}+\sqrt{x+4})\] when you make h small enough (or approaching zero) you get \[= 1/(2(\sqrt{x+4}))\]