Science:Math Exam Resources/Courses/MATH110/December 2013/Question 04 (c)/Solution 1

We want to apply the chain rule. Set $u=\sqrt{x}-1$.

$\begin{array}{rlrl}g(u)& =\sqrt{u}-1& g^{\prime }(u)& =\frac{1}{2\sqrt{u}}\\ u(x)& =\sqrt{x}-1& u^{\prime }(x)& =\frac{1}{2\sqrt{x}}\end{array}$

Combining, we get:

$\begin{array}{rl}{g}^{\prime }(x)& =\frac{1}{2\sqrt{x}}\cdot \frac{1}{2\sqrt{\sqrt{x}-1}}\\ & =\frac{1}{4\sqrt{x}\cdot \sqrt{\sqrt{x}-1}}\\ & =\frac{1}{4\sqrt{x(\sqrt{x}-1)}}\end{array}$