Question

In Exercises \(31-42,\) find \(d y / d x\). $$y=\sqrt{1-\sqrt{x}}$$

Short answer

-\(\frac{1}{4\sqrt{x(1-\sqrt{x})}}\)

Step-by-step solution

Apply the Chain Rule

The outer function is \(\sqrt{v}\) and the inner function is \(1 - u\) where \(u = \sqrt{x}\) Apply chain rule to find the derivative \(d y / d x\). \[y' = \frac{1}{2\sqrt{v}} \cdot (-1) \cdot u'\]

Solve for \(u'\)

Now it is necessary to solve for \(u'\). As \(u = \sqrt{x}\), its derivative, \(u'\), can be calculated using the power rule which results into: \[u' = \frac{1}{2\sqrt{x}}\]

Substitute \(u'\) into the chain rule equation

Substitute \(u'\) from step 2 into the chain rule equation: \[y' = \frac{1}{2\sqrt{1-\sqrt{x}}} \cdot (-1) \cdot \frac{1}{2\sqrt{x}} = -\frac{1}{4\sqrt{x(1-\sqrt{x})}}\]