Table of Contents

## Can a derivative exist at a point where the function does not exist?

When there’s no tangent line and thus no derivative at a sharp corner on a function. See function f in the above figure. Where a function has a vertical inflection point. In this case, the slope is undefined and thus the derivative fails to exist.

**Is the derivative of a function at a point a function?**

A secant line is a line through two points on a curve. secant line that connects two points, and instantaneous velocity corresponds to the slope of a line tangent to the curve. The slope of the tangent line to the graph of a function at a point is called the derivative of the function at that point.

### Does the derivative exist at a hole?

Visually, we can see a point where the derivative (slope of the curve) does not exist (DNE) by looking for “corners” or vertical tangents or “holes” in the graph of the function. What can we say about the derivative of this function at x = -1, 1 and 2?

**How do you find the derivative of a function at a point on a graph?**

Choose a point on the graph to find the value of the derivative at. Draw a straight line tangent to the curve of the graph at this point. Take the slope of this line to find the value of the derivative at your chosen point on the graph.

#### Where the derivative fails to exist?

If there is a discontinuity, a sharp turn, or a vertical tangent at the point, then the derivative does not exist. If you want to prove that the derivative at a certain point doesn’t exist, then you have to find the left-hand and right-hand limits of the point.

**Where are derivatives not taken?**

If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there. So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined.

## Is there a derivative at a point?

The derivative at a point is the limit of slopes of the secant lines or the limit of the difference quotient.

**Where does a derivative exist?**

Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists (i.e., is continuous) on its domain. Thus, a differentiable function is also a continuous function.

### Does derivative zero exist?

f′(x)=limh→0f(x+h)−f(x)h=limh→00h=0. Geometrically speaking, the graph of f:x↦0 is a horizontal line, so its slope at each point is zero, hence its derivative is equal to zero everywhere. From another perspective, f:x↦0 is a constant function, it doesn’t vary, so its rate of change is zero.

**What functions do not have derivatives?**

In the case of functions of one variable it is a function that does not have a finite derivative. For example, the function f(x)=|x| is not differentiable at x=0, though it is differentiable at that point from the left and from the right (i.e. it has finite left and right derivatives at that point).