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) = x^{2} – 4x. On the other hand, if x < 0 then |x| = -x and *f*(x) = -x^{2} – 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) = -x^{2} – 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) = x^{2} – 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**