The function y = x² – 25 is a polynomial function. Polynomial functions are differentiable everywhere on their domain because they are continuous and smooth without any breaks, sharp corners, or vertical tangents.
To express the intervals where the function is differentiable, we can examine the nature of the function. Since it is defined for all real numbers (from negative infinity to positive infinity), we can conclude that:
The function is differentiable for all x values, which in interval notation is represented as:
( -∞, ∞ )