To determine if a function is differentiable at a specific point, you need to follow a few important steps:
- Check Continuity: First, ensure the function is continuous at that point. A function must be continuous in order to be differentiable. You can check continuity by examining the limit of the function as it approaches the point from both sides and comparing it to the function’s value at that point.
- Calculate the Derivative: To see if a function is differentiable at a point, you need to compute the derivative at that point. The derivative can be defined as:
- Evaluate the Limit: If the limit exists, it means the function has a defined slope at that point, indicating that the function is differentiable there. If the limit does not exist, the function is not differentiable at that point.
- Check for Kinks or Corners: Functions can be continuous yet fail to be differentiable at points where the graph has sharp corners or kinks. If there’s a sudden change in direction, the derivative will not exist at that point.
- Consider Endpoints: If the point in question is an endpoint of the function’s domain, make sure to evaluate the one-sided limits, as the derivative is defined in terms of limits approaching from both sides.
f'(a) = lim (h -> 0) [f(a + h) – f(a)] / h
In summary, a function needs to be continuous at a point to be considered differentiable, and you must check the limit of the derivative at that point. Remember to look out for any corners, kinks, and endpoints that may affect differentiability.