When does a function fail to be differentiable?

📌 CONCEPT: A function fails to be differentiable at a point if its derivative does not exist at that point, often due to the presence of a sharp turn or discontinuity in the function's graph.

📐 RULE / FORMULA: The derivative of a function f(x) exists at a point x = a if the limit of the difference quotient [f(x) - f(a)]/(x - a) exists as x approaches a.

💡 WORKED EXAMPLE: For the function f(x) = |x^2 - 4|, consider the point x = 2. To check differentiability, we examine the limit of the difference quotient: lim (x→2) [f(x) - f(2)]/(x - 2). This limit does not exist because the absolute value function has a sharp turn at x = 2, making the function non-differentiable at that point.

⚠️ COMMON MISTAKE: Students often mistake a function's continuity for differentiability, assuming that a function is differentiable at a point if it is continuous there. However, this is not always true, as shown by the example of the absolute value function.