When is function not differentiable

If we substitute these relations into the limits above, we will find that the two definitions of derivative are equivalent. In addition to the notation above, there are several other notations for expressing the derivative of a function. The most common notations are :.

The notation is called Leibniz notation. It is the most commonly used notation after the f' x notation. If we find the derivative for the variable x rather than a value a , we obtain a derivative function with respect to x. With this function, the derivative at any value of x can be determined. By replacing a with x in the limit formula, we can find the derivative function. A function is differentiable at a if f' a exists. It is differentiable on the open interval a , b if it is differentiable at every number in the interval.

If a function is differentiable at a then it is also continuous at a. The contrapositive of this theorem states that if a function is discontinuous at a then it is not differentiable at a.

A function is not differentiable at a if its graph illustrates one of the following cases at a :. A function is not differentiable at a if there is any type of discontinuity at a. This is stated in the contrapositive of the theorem above. The graph to the right illustrates jump discontinuity. A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line.

Since the slope of a vertical line is undefined, the function is not differentiable in this case. A function is not differentiable at a if its graph has a corner or kink at a. As x approaches the corner from the left- and right-hand sides, the function approaches two distinct tangent lines.

Since the function does not approach the same tangent line at the corner from the left- and right-hand sides, the function is not differentiable at that point. The graph to the right illustrates a corner in a graph.

Note: Although a function is not differentiable at a corner, it is still continuous at that point. Taking limits to find the derivative of a function can be very tedious and complicated. The formulas listed below will make differentiating much easier. Each formula is expressed in the regular notation as well as Leibniz notation. Note: The graph of the derivative of a power function will be one degree lower than the graph of the original function. Jim H. Mar 13, There are three ways a function can be non-differentiable.

We'll look at all 3 cases. Case 1 A function in non-differentiable where it is discontinuous. Example 1d description : Piecewise-defined functions my have discontiuities.

Related questions What are non differentiable points for a function? What does differentiable mean for a function? But we can also quickly see that the slope of the curve is different on the left as it is on the right.

This suggests that the instantaneous rate of change is different at the vertex i. We use one-sided limits and our definition of derivative to determine whether or not the slope on the left and right sides are equal.

While the function is continuous, it is not differentiable because the derivative is not continuous everywhere, as seen in the graphs below.

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