Find all values of x for which the function y=(the square root of x)/(5x-2) is differentiable.
In order for $\displaystyle y$ to be differentiable, then $\displaystyle y$ and $\displaystyle y'$ must be defined, or real.
For $\displaystyle y$, the numerator confines anything inside the squareroot to be greater than or equal to zero: $\displaystyle x \geq 0$.
When the denominator equals to zero, we know the function becomes undefined. This occurs when $\displaystyle 5x-2 = 0 \Rightarrow x = \frac{2}{5}$
Thus our conditions are: $\displaystyle x \geq 0$, $\displaystyle x$ not equal to $\displaystyle \frac{2}{5}$
Upon differentiating to $\displaystyle y'$, we can see that the same conditions for $\displaystyle x$ are required for $\displaystyle y'$ to be defined.