Questions: Describe the (x)-values at which (f) is differentiable. (Enter your answer using interval notation.) (y=fracx^2x^2-64)

$Describe the (x)-values at which (f) is differentiable. (Enter your answer using interval notation.) (y=fracx^2x^2-64)$

Transcript text: Describe the $x$-values at which $f$ is differentiable. (Enter your answer using interval notation.) \[ y=\frac{x^{2}}{x^{2}-64} \]

Solution

Solution Steps

Step 1: Determine the domain of $f(x)$

To find the domain of $f(x) = \frac{P(x)}{Q(x)}$, solve $Q(x) \neq 0$. The roots of $Q(x)$ are at x = -8, x = 8, hence $f(x)$ is not defined at these points.

Step 2: Check for differentiability

Rational functions are differentiable wherever they are defined, except at the zeros of $Q(x)$. Therefore, $f(x)$ is not differentiable at x = -8, x = 8.