Final answer:
The function f(x)=(x²e²)/(a²-e⁻ᵗ) is differentiable for all real numbers as long as the denominator a²-e⁻ᵗ is nonzero, implying that a² ≠ e⁻ᵗ.
Step-by-step explanation:
The function given is f(x)= (x²e²)/(a²- e⁻ᵗ). To determine the values of x for which this function is differentiable, we need to look at the conditions where a function might not be differentiable. These typically include points where the function is not continuous, has a sharp corner or cusp, or where the function is asymptotic. In this case, continuity of the function is ensured if the denominator is not equal to zero, which happens when a² = e⁻ᵗ. Assuming 'a' and 'e⁻ᵗ' are constants and without any specific values given for these variables, we can say the function is differentiable as long as the denominator, a² - e⁻ᵗ, is nonzero. Therefore, the function is differentiable for all real numbers provided that a² ≠ e⁻ᵗ.