In mathematics, a left derivative and a right derivative are derivatives (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function.

Let y = f(x) be a function and let a be in the domain of f. The right-hand derivative of f at x = a is the limit. and the left-hand derivative of f at x = a is the limit. The function f is differentiable on the interval I if.

Definition: A formal definition of left and right limits. Let f be a function defined on an interval (b,a\). We say that L=limx→a− f(x) provided that, for every ϵ >0, there exists a δ>0 such that for all x, 0<(a−x)<δ⟹|f(x)−L|<ϵ Similarly, if f is a function defined on an interval (a,b\).

LHL - **Left Hepatic Lobe**.

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- Observe the points where the given function can be non-differentiable.
- Check right hand limit Derivative and left hand derivative at those points.
- If LHD = RHD ,the function is differentiable at those points.
- If LHD = RHD ,the function is not differentiable at those points.

Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as. **f′(a−)=h→0+limh**f(a)−f(a−h)=h→0−limhf(a)−f(a−h)=x→a+lima−xf(a)−f(x) respectively.

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.

To determine if a left-hand limit exists, we observe the branch of the graph to the left of x = a \displaystyle x=a x=a, but near x = a \displaystyle x=a x=a. This is where x < a \displaystyle x<a x<a. We see that the outputs are getting close to some real number L so there is a left-hand limit.

What are One-Sided Derivatives? **When you differentiate or find a derivative of a function either from the left or from the right**, then such derivative is known as a one-sided derivative.

The right-hand derivative of f is defined as **the right-hand limit**: f′+(x)=limh→0+f(x+h)−f(x)h. If the right-hand derivative exists, then f is said to be right-hand differentiable at x.

In mathematics, a left derivative and a right derivative are derivatives (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function.

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Dated : 20-Jul-2022

Category : Education