ONE-SIDED DERIVATIVES

Just as the concept of one-sided limits in the study of limits of functions, we may find the concept of one-sided derivatives when we learn calculus. In the article The Limit of A Function at A Point, it is claimed that the limit of a function at a point exists if and only if the value of its left-hand limit is equal to the value of its right-hand limit. Similarly, the derivative of a function at a point is guaranteed to exist if and only if the value of its left-hand derivative is equal to the value of its right-hand derivative. The left- and right-hand derivatives are referred to as one-sided derivatives.

 

Left-hand Derivatives

Let f be a function defined on an interval (a,c]. The left-hand derivative of f at x = c, denoted by , is defined as follows.

……………………………………………………………………………………………………………………………………………………… (1)

if the limit exists. Equivalently, (1) can be expressed as:

………………………………………………………………………………………………………………………………………………. (1a)

 

Right-hand Derivatives

Let f be a function defined on an interval [c,b). The right-hand derivative of f at x = c, denoted by , is defined as follows.

……………………………………………………………………………………………………………………………………………………. (2)

if the limit exists. Equivalently, (2) can be expressed as:

…………………………………………………………………………………………………………………………………………….. (2a)

 

The relationship between the one-sided derivatives of a function at a point and the differentiability of the function at the point is stated by the following theorem.

 

Theorem

Let f be a real-valued function defined on an open interval (a,b) and c ∈ (a, b).

f ‘(c) exists if and only if f ‘_– (c) = f ‘₊(c).

 

Example 1

Let f be a function from to defined by f(x) = x⋅|x| – 4x. Find both one-sided derivatives of f at x = 0. Is f differentiable at x = 0?

Answer

If x ≥ 0 then |x| = x and consequently f(x) can be expressed as f(x) = x² – 4x. On the other hand, if x < 0 then |x| = -x and f(x) = -x² – 4x. Thus, f(x) can be expressed as:

By (1), the left-hand derivative of f at x = 0 is computed as follows.

…………………………………………………………………………………………………………………………………………………… (*)

As f(0) = 0, (*) can be expressed as:

………………………………………………………………………………………………………………… (**)

Since x approaches 0 from the left, the value of x is always negative. Hence f(x) = -x² – 4x. Substituting this into (**), we get:

By (2), the right-hand derivative of f at x = 0 is computed as follows.

……………………………………………………………………………………………………………. (***)

Since x approaches 0 from the right, the value of x is always positive. Hence f(x) = x² – 4x. Substituting this into (***), we get:

Since f ‘_–(0) = f ‘₊(0), by the theorem above, we conclude that f is differentiable at x = 0. (See Figure 1.)

Figure 1

 

Example 2

Let g be a function from to defined by . Find both one-sided derivatives of g at x = 0. Is g differentiable at x = 0?

Answer

Note that g(x) can be expressed as:

By (1), the left-hand derivative of g at x = 0 is computed as follows.

  ………………………………………………………………………………………………………………………………………………. (+)

As g(0) = 0, (+) can be expressed as:

         …………………………………………………………………………………………………………………………………………… (++)

Since x approaches 0 from the left, the value of x is always negative so g(x) = 0. Substituting this into (++), we get:

By (2), the right-hand derivative of g at x = 0 is calculated as follows.

……………………………………………………………………………………………………………. (+++)

Since x approach 0 from the right, the value of x is always positive.  Hence g(x) = x. Substituting this into (+++), we get:

.

As g ‘_–(0) ≠ g ‘₊(0), by the theorem above, we conclude that g is not differentiable at x = 0. (See Figure 2.)

Figure 2