Differentiable Function: Meaning, Formulas and Examples
03.10.2022 • 7 min read
Subject Matter Expert
In this article, we’ll discuss the definition of differentiable. Using graphed examples, we’ll learn how to spot where a function is non-differentiable. Then, we’ll review the difference between differentiable and continuous.
What does differentiable mean? If a function is differentiable, its derivative exists at every point in its domain. If a function is differentiable at a point x, the limit of the average rate of change of f over the interval [x,x+Δx] as Δx approaches 0 exists.
Let’s unpack what the limit means. The function inside this limit probably looks familiar. The average rate of change over an interval, otherwise known as the difference quotient, measures a function’s slope between two points.
This slope value represents how fast a function’s output values (y-values) change with respect to its input (x-values). The delta symbol Δx is used to represent the value that a variable changes by.
The formula for the average rate of change of the function f over the interval [a,b] is below.
Average Rate of Change=ΔxΔy=x2−x1y2−y1=b−af(b)−f(a)
The difference quotient is also commonly represented like this:
Δxf(x+Δx)−f(x) or hf(x+h)−f(x) where h=Δx
Here, the delta symbol Δx represents the value that x changes by. When we make Δx approach 0 in the limit below, we can find the derivative of a function or instantaneous rate of change. This value also represents the slope of the tangent line.
If this limit exists, then L is the derivative of f(x). This is denoted by f’(x) or dxdy.
Dr. Hannah Fry dives more into what a derivative is:
Let’s look at some differentiable examples of functions.
All polynomial functions are differentiable everywhere, as are constant functions. A rational function is differentiable except at the x-value that makes its denominator 0.
What Makes a Function Non-Differentiable?
Now, let’s learn how to find where a function is not differentiable. If a function has any discontinuities, it is not differentiable at those points. In order to be differentiable, a function must be continuous. The output value must be defined for each input value.
Second, the limit as Δx approaches the difference quotient must exist in order for a function to be differentiable at a point x. The limit does not exist if the limit as Δx approaches 0 from the left does not equal the limit as Δx approaches 0 from the right.
This might happen if a function is not continuous at x, or if the function’s graph has a corner point, cusp, or vertical tangent.
Knowing what corner points, cusps, vertical tangents, and discontinuities look like on a graph can help you pinpoint where a function is not differentiable. Let’s examine some non-differentiable graph examples below.
The function f(x)=cos−1(cos(x)) is an example of a function with corner points, such as at x=π. A corner point looks like two linear sections of a function that meet at a sharp point.
The slope of the tangent line to the left of a corner point is different from the slope to the right of the corner point. Because of this, a function’s slope is not defined at a corner point, so its derivative cannot be calculated there.
The function f(x)=2(x−1)(32) is an example of a function with a cusp at x=1. A cusp looks like two curves that meet at a sharp point.
The slopes of the tangent lines to the left of the cusp approach −∞, while the slopes of the tangent lines to the right of the cusp approach +∞. Because of this, a function’s slope is not defined at a cusp, so its derivative cannot be calculated there.
The function f(x)=3x is an example of a function with a vertical tangent. At x=0, the slope of the tangent line approaches infinity.
A function f has a vertical tangent at x if f is continuous at x and if the slope of the tangent line at x approaches either negative infinity or positive infinity.
The function f(x)=(x2)1+u(x−2)+u(x−9), where u represents the unit-step function, is an example of a function with a discontinuity. For example, there are jump continuities at x=2 and x=9.
Since the limit as x approaches these points from the left does not equal the limit as x approaches these points from the right, f(x) is not differentiable at these points.
What is the Difference Between Differentiable and Continuous?
A differentiable function must be continuous. However, the reverse is not necessarily true. It’s possible for a function to be continuous but not differentiable. (If needed, you can review our full guide on continuous functions.)
Let’s examine what it means to be a differentiable versus continuous function. For example, consider the absolute value function f(x)=∣x∣ below.
This function is continuous everywhere because we can draw its curve without ever lifting a hand. Its curve has no holes, breaks, jumps, or vertical asymptotes. However, at x=0, the function is not differentiable.
How can we tell this function is not differentiable?
We know it’s non-differentiable because there’s a corner point at x=0. This makes it impossible to draw the tangent line to f(x)=∣x∣ at x=0. More precisely, the absolute value function fails the limit definition of differentiability at x=0.
Let’s verify that the absolute value function fails the limit definition of differentiability at x=0 by plugging f(x)=∣x∣ at x=0 into the limit definition of a derivative formula. So that we don’t confuse x and Δx, we’ll substitute the variable h for Δx.
We are calculating h→0limhf(x+h)−f(x) for f(x)=∣x∣ at x=0.
Here’s what we get:
Plugging in x=0, we have:
Let’s stop and take a closer look at the function h∣h∣, which can be written as a piecewise function. This piecewise function represents f’(x), the derivative of our function f(x)=∣x∣.
We can use this piecewise function to finish evaluating our limit and to understand why f’(x) is non-differentiable at x=0.
First, let’s take the limit as h approaches 0 from the right. Imagine h as a slightly positive value, so that h>0.
Looking at our piecewise function, we can plug in h∣h∣=hh=1 for h>0. Remember that the limit as h approaches a constant value is simply the constant value itself.
Now, let’s take the limit as h approaches 0 from the left. Imagine h as a slightly negative value, so that h < 0. Looking at our piecewise function, we can plug in for h < 0 the following:
h∣h∣ = h−h=−1
Here's what it'll look like:
Notice that h→0+lim=h→0−limh∣h∣, since 1=−1. In order for a limit to exist, the right-handed limit must equal the left-handed limit.
On our graph of f’(x)=x∣x∣ below, this looks like a jump discontinuity.
This means that h→0limhf(x+h)−f(x) for f(x)=∣x∣ at x=0 does not exist. Thus, f(x)=∣x∣ is not differentiable at x=0.
So, although f(x)=∣x∣ is continuous everywhere, it is not differentiable at x=0. The same is true for many other functions, so make sure you understand the difference.