How to Determine if a Function is Differentiable at a Point

To determine if a function is differentiable at a specific point, you need to check a few conditions:

1. Continuity: First, the function must be continuous at that point. If there is any kind of discontinuity, like a jump or an asymptote, the function cannot be differentiable there.
2. Existence of the Derivative: You need to confirm that the derivative exists at that point. This is done by calculating the limit of the difference quotient as the point approaches from both the left and the right. Specifically, you want to find:

f'(a) = lim (h -> 0) [f(a + h) - f(a)] / h

3. If the limit exists and is the same from both sides, then the function is differentiable at that point.

Additionally, if the left-hand limit and the right-hand limit of the derivative are not equal, then the function is not differentiable at that point, even if it is continuous. An intuitive way to think of differentiability is that the function has a tangent at that point; if it has any sharp corners or vertical tangents, it won’t be differentiable there.