Hi Renee,

If $x = a$ is in the domain of the function $y = f^{\;\prime}$ the, first of all $x = a$ must be in the domain of $y = f(x).$ Think of the derivative of $y = f(x)$ at $x = a$ as the slope of the tangent to $y = f(x)$ at $x = a.$ For the tangent to $y = f(x)$ at $x = a$ to exist there must be a point $(a, f(a))$ on the the graph and hence $f(a)$ must exist. There might however be points where the function $f(x)$ exists but $f^{\;\prime}(x)$ does not exist.

Consider your function $f(x) = \sqrt{8 - x}.$ The domain of $f(x)$ is all $x$ so that $8 - x \leq 0$ that is the domain of $f(x)$ is all $x$ so that $x \leq 8.$ The derivative of $f(x)$ is

$$f^{\;\prime}(x) = \frac{-1}{2 \sqrt{8 - x}}.$$

In this function $x$ can not be 8 or the denominator is zero and hence the domain of $f^{\;\prime}(x)$ is all $x$ so that $x \lt 8.$

Penny