# Differentiable

How to find if the given function is differentiable: definition, relationship between continuous and differentiable, examples, and their solutions.

## Definition

A function is differentiable

if the left-hand derivative and the right-hand derivative

at a point are equal.

Derivative

This makes sense

because the limit value is defined

if the left-side limit and the right-side limit are equal.

One-sided limits

In plain words,

if a function is differential at a point,

its graph changes smoothly at that point.

(no sharp point made by sudden change of its slope)

## Continuous and Differentiable

A differentiable function is always continuous.

The graph of a differential function changes smoothly.

This implies that the graph is always continuous.

But a continuous function is not always differentiable.

The graph of a continuous function can also be sharp.

At the sharp point,

the function is not differentiable (= not smooth).

## Example 1

Recall that

a differentiable function is always continuous.

So first show that *f*(*x*) is continuous at *x* = 1.

Then show that *f*(*x*) is differentiable at *x* = 1.

The left-hand limit of *f*(*x*) is 3.

One-sided limits

The right-hand limit of *f*(*x*) is also 3.

And *f*(1) = 3.

The left-hand limit, the right hand limit,

and the function value are all 3.

So *f*(*x*) is continuous at *x* = 1.

Next, see if *f*(*x*) is differentiable at *x* = 1.

Find the left-hand derivative

by using the definition of the derivative.

(1 + *h*)^{2} = 1^{2} + 2⋅1⋅*h* + *h*^{2}

Square of a sum

Cancel the dark gray terms.

Divide the numerator and the denominator by *h*.

So the left-hand derivative is 2.

Find the right-hand derivative

by using the definition of the derivative.

(1 + *h*)^{2} = 1^{2} + 2⋅1⋅*h* + *h*^{2}

Square of a sum

Cancel the dark gray terms.

Add the *h* terms. (gray)

Divide the numerator and the denominator by *h*.

So the right-hand derivative is 2.

The left-hand derivative and the right-hand derivative

are equal: 2.

So *f*(*x*) is differentiable at *x* = 1.

Let's see what this means.

The left-hand derivative and the right-hand derivative

are equal: 2.

So the slope of the tangent line at *x* = 1 is 2,

whether *x* comes from the left or the right.

So the graph of *f*(*x*) is smooth at *x* = 1.

## Example 2

*f*(*x*) = |*x*| is a piecewise function.

So write *f*(*x*) by its range:*f*(*x*) = -*x* at (*x* < 0)

= *x* at (*x* ≥ 0).

First, show that *f*(*x*) is continuous at *x* = 0.

The left-hand limit of *f*(*x*) is 0.

One-sided limits

The right-hand limit of *f*(*x*) is also 0.

And *f*(0) = 0.

The left-hand limit, the right hand limit,

and the function value are all 0.

So *f*(*x*) is continuous at *x* = 0.

Next, see if *f*(*x*) is differentiable at *x* = 1.

Find the left-hand derivative

by using the definition of the derivative: -1.

Find the right-hand derivative

by using the definition of the derivative: 1.

The left-hand derivative (-1)

and the right-hand derivative (1)

are not equal.

So *f*(*x*) is not differentiable at *x* = 0.

Let's see what this means.

The left-hand derivative (-1)

and the right-hand derivative (1)

are not equal.

This means

the left-hand slope (-1)

and the right-had slope (1)

are not equal.

So the slope of the tangent line at *x* = 1

cannot be determined.

And see that the graph of *f*(*x*) is sharp at *x* = 0.

The sharp point is formed

because of the sudden change of the slope: -1 → 1.