Why corners are not differentiable

Connect and share knowledge within a single location that is structured and easy to search. I am learning about differentiability of functions and came to know that a function at sharp point is not differentiable. For eg. How sharp point makes these limits to evaluate different? Be very careful, if you use it to disprove differentiability. But again, be careful: differentiability is a mathematical idea. The best way to understand it, is to understand it mathematically , according to the definition.

Everything else may be misleading. That is, up close, the function looks like a straight line. It's not differentiable because you can draw infinitely many tangents that touch the point of turning I may be wrong I'm just a high school student. Because in order to calculate a derivative you need to take the difference of the point in question and the next point on the closest possible interval.

This means that you must take the difference from the right and from the left. When you encounter a kink if you take the difference of the closest point from the left and from the right you will get reciprocal answers. Therefor it is impossible to determine the rate of change at the point in question.

Hence a turning point that is curved IS differentiable, but this 'cusp' is not. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams?

The function g x is substituted for x into the function f x. Often, a function can be written as a composition of several different combinations of functions. The chain rule allows us to find the derivative of composite functions.

The limits below are required for proving the derivatives of trigonometric functions. These limits and the derivatives of the trigonometric functions will be proven in your calculus lectures. Here, they are simply stated. Note: These limits are used often when solving trigonometric limit problems. Try to remember them and the conditions under which they hold. Note: The derivatives of the co-functions cosine, cosecant and cotangent have a "-" sign at the beginning.

This is a helpful way to remember the signs when taking the derivatives of trigonometric functions. The method of implicit differentiation allows us to find the derivative of an implicit function. It allows us to differentiate y without solving the equation explicitly. We can simply differentiate both sides of the equation and then solve for y '. When differentiating a term with y , remember that y is a function of x. The term is a composition of functions, so we use the chain rule to differentiate.

For example, if you were to differentiate the term 3 y 4 it would become 12 y 3 y '. Note: For a more concrete demonstration of how to differentiate implicit functions, see example 14 below.

Earlier in the derivatives tutorial, we saw that the derivative of a differentiable function is a function itself. If the derivative f' is differentiable, we can take the derivative of it as well. The new function, f'' is called the second derivative of f. If we continue to take the derivative of a function, we can find several higher derivatives. In general, f n is called the nth derivative of f. Note: Recall that when working with motion application problems, the velocity of the particle is the first derivative of the displacement function.

The acceleration of the particle is the derivative of the velocity function, or equivalently, the second derivative of the displacement function. It is often easy to calculate the exact value of a function at a point a , but rather difficult to compute values near a. We can find an approximate value of the function at points near a by using the tangent line to the curve at a. For more practice with the concepts covered in the derivatives tutorial, visit the Derivatives Problems page at the link below.

The solutions to the problems will be posted after the derivatives chapter is covered in your calculus course. Do one-sided limits always exist? Can a graph be continuous at a corner? Why do derivatives not exist at sharp corners? Can derivatives be zero? Are endpoints critical points? What is the difference between a corner and a cusp? What does it mean when the tangent line is vertical?

Why are cusps and corners not differentiable? Are corners and cusps differentiable? Do limits exist at cusps?

Are functions differentiable at sharp corners? Can a function be differentiable at a hole? What kinds of functions are not differentiable? Why are vertical tangents not differentiable? How do you know if a tangent line is vertical? How do you know if a function is not differentiable? How do you know if a function is continuous or differentiable? Can a function be differentiable and not continuous?