1. Finding Differentiable Values

Find all values of x for which the function y=(the square root of x)/(5x-2) is differentiable.

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

Upon differentiating to $y'$, we can see that the same conditions for $x$ are required for $y'$ to be defined.