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Calculus

# Differentiable Function: Meaning, Formulas and Examples

## 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.

## 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 $x$, the limit of the average rate of change of $f$ over the interval $[x, x +\Delta{x}]$ as $\Delta{x}$ approaches 0 exists.

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

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 $\Delta{x}$ 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.

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

### Difference Quotient

The difference quotient is also commonly represented like this:

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

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

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

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

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

### Differentiable Examples

Let’s look at some differentiable examples of functions.

• $f(x) = 4x^3 - 7x$

• $f(x) = 12$

• $f(x) = \sin{(x)}$

• $f(x) = \cos{(x)}$

• $f(x) = e^x$

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 $\Delta{x}$ 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 $\Delta{x}$ approaches 0 from the left does not equal the limit as $\Delta{x}$ 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.

### Corner

The function $f(x)=\cos^{-1}\left(\cos\left(x\right)\right)$ is an example of a function with corner points, such as at $x = \pi$. 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.

### Cusp

The function $f(x) = 2\left(x-1\right)^{\left(\frac{2}{3}\right)}$ 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 $-\infty$, 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 $f(x)=\sqrt{x}$ 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.

### Discontinuity

The function $f(x)=\frac{1}{(x^{2})}+u\left(x-2\right)+u\left(x-9\right)$, 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 $f(x) = \vert x \vert$ 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 $x = 0$, 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 $f(x) = \vert x \vert$ 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 $f(x) = \vert x \vert$ at $x = 0$ into the limit definition of a derivative formula. So that we don’t confuse $x$ and $\Delta{x}$, we’ll substitute the variable $h$ for $\Delta{x}$.

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

Here’s what we get:

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

Plugging in $x = 0$, we have:

$\mathop {\lim }\limits_{h \to 0} \frac{\vert 0 + h \vert - \vert 0 \vert}{h} = \mathop {\lim }\limits_{h \to 0} \frac{\vert h \vert}{h}$

Let’s stop and take a closer look at the function $\frac{\vert h \vert}{h}$, which can be written as a piecewise function. This piecewise function represents $f’(x)$, the derivative of our function $f(x) = \vert x \vert$. 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 $\frac{\vert h \vert}{h} = \frac{h}{h} = 1$ for $h > 0$. Remember that the limit as $h$ approaches a constant value is simply the constant value itself.

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

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:

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

Here's what it'll look like:

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

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

On our graph of $f’(x) = \frac{\vert x \vert}{x}$ below, this looks like a jump discontinuity. This means that $\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h } \right) - f\left( x \right)}}{h }$ for $f(x) = \vert x \vert$ at $x = 0$ does not exist. Thus, $f(x) = \vert x \vert$ is not differentiable at $x = 0$.

So, although $f(x) = \vert x \vert$ 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.

## 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

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

### Power Rule

$\frac{d}{dx}(x^n) = nx^{n-1}$

### Chain Rule

$\frac{d}{dx}f(g(x)) = f’(g(x))g’(x)$

### Product Rule

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

### Quotient Rule

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

### Sum/Difference Rule

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

### Trigonometry Rules

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

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

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

### Logarithmic and Exponential Rules

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

• $\frac{d}{dx}(e^x) = e^x$

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

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