haha1234 said:
pasmith said:
The concept of differentiability is a mathematical term used to describe the smoothness of a function. A function is considered differentiable at a point if it has a well-defined derivative at that point, meaning that the function is continuous and has a defined slope at that point. In other words, the function is considered differentiable if it has a unique tangent line at that point.
To prove that a function is differentiable at a point, you can use the definition of differentiability. This involves showing that the function is continuous at that point and that the limit of the difference quotient (the slope of the secant line between two points on the function) exists and is equal to the slope of the tangent line at that point. You can also use the rules of differentiation to show that the function is differentiable at that point.
Yes, it is possible for a function to be differentiable at some points and not others. This is because differentiability is a local property - it only applies to a specific point on the function. A function may be differentiable at one point but not at another if it has a sharp corner or a discontinuity at that point.
Differentiability and continuity are related concepts, but they are not the same. A function is considered continuous if it is unbroken and has no holes or gaps on its graph. Differentiability, on the other hand, refers to the smoothness of a function at a specific point. A function can be continuous at a point but not differentiable, and vice versa.
Yes, there are some shortcuts or criteria that can help you determine whether a function is differentiable at a point. For example, if a function is composed of basic elementary functions (such as polynomials, trigonometric functions, and exponential functions), it is always differentiable at every point in its domain. Additionally, if a function is continuous and has a defined slope at a point, it is also differentiable at that point. However, these are not foolproof methods and it is always best to use the definition of differentiability to prove it rigorously.