spiked structures representing what does differentiable mean

Calculus

Differentiable Function: Meaning, Formulas and Examples

03.10.2022 • 7 min read

Rachel McLean

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.

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In This Article

  1. What is Differentiable?

  2. What Makes a Function Non-Differentiable?

  3. What is the Difference Between Differentiable and Continuous?

  4. Common Derivative Formulas

What is Differentiable?

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 xx, the limit of the average rate of change of ff over the interval [x,x+Δx][x, x +\Delta{x}] as Δx\Delta{x} approaches 0 exists.

f(x)=limΔx0ΔyΔx=limΔx0f(x+Δx)f(x)Δx=Lf’(x) = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\Delta{y}}{\Delta{x}} = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} } = L

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\Delta{x} is used to represent the value that a variable changes by.

The formula for the average rate of change of the function ff over the interval [a,b][a, b] is below.

Average Rate of Change=ΔyΔx=y2y1x2x1=f(b)f(a)ba\text{Average Rate of Change} = \frac{\Delta{y}}{\Delta{x}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(b)-f(a)}{b-a}

Difference Quotient

The difference quotient is also commonly represented like this:

f(x+Δx)f(x)Δx\frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} } or f(x+h)f(x)h\frac{{f\left( {x + h } \right) - f\left( x \right)}}{h } where h=Δxh = \Delta{x}

Here, the delta symbol Δx\Delta{x} represents the value that xx changes by. When we make Δx\Delta{x} 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.

f(x)=limΔx0ΔyΔx=limΔx0f(x+Δx)f(x)Δx=Lf’(x) = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\Delta{y}}{\Delta{x}} = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} } = L

If this limit exists, then LL is the derivative of f(x)f(x). This is denoted by f(x)f’(x) or dydx\frac{dy}{dx}.

Dr. Hannah Fry dives more into what a derivative is:

Differentiable Examples

Let’s look at some differentiable examples of functions.

  • f(x)=4x37xf(x) = 4x^3 - 7x

  • f(x)=12f(x) = 12

  • f(x)=sin(x)f(x) = \sin{(x)}

  • f(x)=cos(x)f(x) = \cos{(x)}

  • f(x)=exf(x) = e^x

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\Delta{x} approaches the difference quotient must exist in order for a function to be differentiable at a point xx. The limit does not exist if the limit as Δx\Delta{x} approaches 0 from the left does not equal the limit as Δx\Delta{x} approaches 0 from the right.

This might happen if a function is not continuous at xx, 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.

Corner

The function f(x)=cos1(cos(x))f(x)=\cos^{-1}\left(\cos\left(x\right)\right) is an example of a function with corner points, such as at x=πx = \pi. A corner point looks like two linear sections of a function that meet at a sharp point.

example of a function with corner points

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.

Cusp

The function f(x)=2(x1)(23)f(x) = 2\left(x-1\right)^{\left(\frac{2}{3}\right)} is an example of a function with a cusp at x=1x = 1. A cusp looks like two curves that meet at a sharp point.

example of a function with a cusp

The slopes of the tangent lines to the left of the cusp approach -\infty, while the slopes of the tangent lines to the right of the cusp approach ++\infty. Because of this, a function’s slope is not defined at a cusp, so its derivative cannot be calculated there.

Vertical Tangent

The function f(x)=x3f(x)=\sqrt[3]{x} is an example of a function with a vertical tangent. At x=0x = 0, the slope of the tangent line approaches infinity.

example of a function with a vertical tangent

A function ff has a vertical tangent at xx if ff is continuous at xx and if the slope of the tangent line at xx approaches either negative infinity or positive infinity.

Discontinuity

The function f(x)=1(x2)+u(x2)+u(x9)f(x)=\frac{1}{(x^{2})}+u\left(x-2\right)+u\left(x-9\right), where uu represents the unit-step function, is an example of a function with a discontinuity. For example, there are jump continuities at x=2x = 2 and x=9x = 9.

example of a function with a discontinuity

Since the limit as xx approaches these points from the left does not equal the limit as xx approaches these points from the right, f(x)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)=xf(x) = \vert x \vert 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=0x = 0, the function is not differentiable.

the absolute value function example that is continuous everywhere, because we can draw its curve

How can we tell this function is not differentiable?

We know it’s non-differentiable because there’s a corner point at x=0x = 0. This makes it impossible to draw the tangent line to f(x)=xf(x) = \vert x \vert at x=0x = 0. More precisely, the absolute value function fails the limit definition of differentiability at x=0x = 0.

Let’s verify that the absolute value function fails the limit definition of differentiability at x=0x = 0 by plugging f(x)=xf(x) = \vert x \vert at x=0x = 0 into the limit definition of a derivative formula. So that we don’t confuse xx and Δx\Delta{x}, we’ll substitute the variable hh for Δx\Delta{x}.

We are calculating limh0f(x+h)f(x)h\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h } \right) - f\left( x \right)}}{h } for f(x)=xf(x) = \vert x \vert at x=0x = 0.

Here’s what we get:

limh0x+hxh\mathop {\lim }\limits_{h \to 0} \frac{\vert x + h \vert - \vert x \vert}{h}

Plugging in x=0x = 0, we have:

limh00+h0h=limh0hh\mathop {\lim }\limits_{h \to 0} \frac{\vert 0 + h \vert - \vert 0 \vert}{h} = \mathop {\lim }\limits_{h \to 0} \frac{\vert h \vert}{h}

Let’s stop and take a closer look at the function hh\frac{\vert h \vert}{h}, which can be written as a piecewise function. This piecewise function represents f(x)f’(x), the derivative of our function f(x)=xf(x) = \vert x \vert.

 example of a piecewise function

We can use this piecewise function to finish evaluating our limit and to understand why f(x)f’(x) is non-differentiable at x=0x = 0.

First, let’s take the limit as hh approaches 0 from the right. Imagine hh as a slightly positive value, so that h>0h >0.

Looking at our piecewise function, we can plug in hh=hh=1\frac{\vert h \vert}{h} = \frac{h}{h} = 1 for h>0h > 0. Remember that the limit as hh approaches a constant value is simply the constant value itself.

limh0+hh=limh0+1=1\mathop {\lim }\limits_{h \to 0^+} \frac{\vert h \vert}{h} = \mathop {\lim }\limits_{h \to 0^+} 1 = 1

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:

hh\frac{\vert h \vert}{h} = hh=1 \frac{-h}{h} = -1

Here's what it'll look like:

limh0hh=limh0+=1\mathop {\lim }\limits_{h \to 0^-} \frac{\vert h \vert}{h} = \mathop {\lim }\limits_{h \to 0^+} = -1

Notice that limh0+limh0hh\mathop {\lim }\limits_{h \to 0^+} \not = \mathop {\lim }\limits_{h \to 0^-} \frac{\vert h \vert}{h}, since 111 \not = -1. In order for a limit to exist, the right-handed limit must equal the left-handed limit.

On our graph of f(x)=xxf’(x) = \frac{\vert x \vert}{x} below, this looks like a jump discontinuity.

example of a jump discontinuity

This means that limh0f(x+h)f(x)h\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h } \right) - f\left( x \right)}}{h } for f(x)=xf(x) = \vert x \vert at x=0x = 0 does not exist. Thus, f(x)=xf(x) = \vert x \vert is not differentiable at x=0x = 0.

So, although f(x)=xf(x) = \vert x \vert is continuous everywhere, it is not differentiable at x=0x = 0. The same is true for many other functions, so make sure you understand the difference.

Common Derivative Formulas

Once you’ve learned how to determine if a function is differentiable, you can start to become familiar with the most common derivative formulas and their rules.

Here is a list of the most useful derivative rules to memorize:

Constant Rule

ddxc=0\frac{d}{dx}c = 0

Power Rule

ddx(xn)=nxn1\frac{d}{dx}(x^n) = nx^{n-1}

Chain Rule

ddxf(g(x))=f(g(x))g(x)\frac{d}{dx}f(g(x)) = f’(g(x))g’(x)

Product Rule

ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x)\frac{d}{dx}[f(x) \cdot g(x)] = f’(x) \cdot g(x) + f(x)\cdot g’(x)

Quotient Rule

ddx[f(x)g(x)]=g(x)f(x)f(x)g(x)(g(x))2\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}

Sum/Difference Rule

ddx[f(x)±g(x)]=f(x)±g(x)\frac{d}{dx}[f(x) \pm g(x)] = f’(x) \pm g’(x)

Trigonometry Rules

  • ddx(sin(x))=cos(x)\frac{d}{dx}(\sin{(x)}) = \cos{(x)}

  • ddx(cos(x))=sin(x)\frac{d}{dx}(\cos{(x)}) = -\sin{(x)}

  • ddx(tan(x))=sec2(x)\frac{d}{dx}(\tan{(x)}) = \sec ^2 (x)

Logarithmic and Exponential Rules

  • ddx(lnx)=1x\frac{d}{dx} (\ln{x}) = \frac{1}{x}

  • ddx(ex)=ex\frac{d}{dx}(e^x) = e^x

Dr. Tim Chartier discusses the Product and Quotient derivative rules more in depth:

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