Question

How do you find the derivative of $y=\sqrt{x-1}$?

Answer 1

In this problem we have to use the Power Rule and the Chain Rule.
We begin by converting the radical(square root) to it exponential form.
$y=\sqrt{x-1}=(x-1{)}^{\frac{1}{2}}$
Apply the Chain Rule
$y^{\prime }=\frac{1}{2}(x-1{)}^{\frac{1}{2}-1}.(1)$
$y^{\prime }=\frac{1}{2}(x-1{)}^{\frac{1}{2}-\frac{2}{2}}$
$y^{\prime }=\frac{1}{2}(x-1{)}^{-\frac{1}{2}}$
Convert negative exponents to positive exponents
$y^{\prime }=\frac{1}{2(x-1{)}^{\frac{1}{2}}}$
Convert positive exponent to radical form
$y^{\prime }=\frac{1}{2\sqrt{x-1}}$