A differentiable function is a function that has a well-defined derivative at every point in its domain. This means that the slope of the function at any point can be determined by taking the limit of the slope of the function at that point as the distance between two points on the function approaches zero.
To determine if a function is differentiable at a specific point, we can use the definition of a derivative to calculate the limit of the slope of the function at that point. If the limit exists and is a finite value, then the function is differentiable at that point.
The common types of points where a function is not differentiable include sharp corners, cusps, and discontinuities. These points occur when the function has a sudden change in slope or direction, making it impossible to calculate the derivative using the limit definition.
To find the points where a function is not differentiable, we can look for any points where the function has a sharp corner, cusp, or discontinuity. We can also use the limit definition of the derivative to determine if the function is differentiable at a specific point. If the limit does not exist or is infinite, then the function is not differentiable at that point.
It is important to identify the points where a function is not differentiable because these points can help us understand the behavior of the function. They can also help us identify any potential errors in our calculations or assumptions about the function. Additionally, these points can be useful in determining the continuity and smoothness of a function, which are important concepts in mathematics and physics.