Definition of Differentiation

Sarah Miller

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Study Guide Overview

This study guide covers differentiability and its relationship with continuity. It explains how to determine if a function is differentiable using the limit definition of the derivative, and how to identify points of non-differentiability (e.g., cusps, vertical tangents, discontinuities). It also includes worked examples, practice questions, and a glossary of key terms like derivative, limit, and asymptote.

#Differentiability & Continuity

#Table of Contents

Differentiability
Relationship Between Differentiability and Continuity
Showing Non-Differentiability
Points of Non-Differentiability on Graphs
Worked Example
Practice Questions
Glossary
Summary and Key Takeaways

#Differentiability

#When is a Function Differentiable?

The derivative of a function is defined as ${f}^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}$
The derivative only exists if this limit exists.
The derivative, if it exists, is itself a function.
If a point is not in the domain of $f(x)$ , it cannot be in the domain of $f^{\prime}(x)$ .
For instance, $f(x)=\frac{1}{x}$ is not differentiable at $x=0$ due to the vertical asymptote.
A differentiable function has its derivative exist at each point in its domain. A function with points at which it is not differentiable can still be made a differentiable function by appropriately restricting its domain.

#Relationship Between Differentiability and Continuity

A function must be continuous at a point to be differentiable at that point.

Key Concept

If a function is differentiable at a point, then it is continuous at that point. Conversely, if a function is not continuous at a point, it is not differentiable at that point.

However, a function being continuous at a point does not necessarily mean it is differentiable at that point.
For instance, $f(x) = |x|$ is continuous at $x=0$ , but it is not differentiable at $x=0$ .

Exam Tip

Exam questions often state that a function is differentiable and expect you to know (and use the fact) that this automatically means the function is continuous as well.

Exam Tip

If a function is said to be twice differentiable, this means that the function's derivative is also continuous!

#Showing Non-Differentiability

#How Can I Show a Function is Not Differentiable at a Point?

If a function is not continuous at a point, then it is not differentiable at that point.

Key Concept

That's the easiest way to show a function is not differentiable at a point!

If a function is continuous at a point, then to show it is not differentiable, you need to go back to the limit definition of the derivative. ${f}^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}$
Recall that the limit of a function does not exist if the function inside the limit:
- is unbounded at the point in question,
- oscillates near the point in question,
- or has unequal one-sided limits at the point in question.

Consider the function

f(x) = |x|

x=0

To show why it is not differentiable, consider: ${f}^{\prime}(0) = \lim_{{h \to 0}} \frac{f(0+h) - f(0)}{h} = \lim_{{h \to 0}} \frac{|h|}{h}$

For $h < 0$ , $\frac{|h|}{h} = \frac{-h}{h} = -1$ . $\lim_{{h \to 0^-}} \frac{|h|}{h} = -1$
For $h > 0$ , $\frac{|h|}{h} = \frac{h}{h} = 1$ . $\lim_{{h \to 0^+}} \frac{|h|}{h} = 1$

The one-sided limits do not agree, therefore the limit does not exist at $x=0$ . Hence, $f(x) = |x|$ is not differentiable at $x=0$ .

Exam Tip

If you have to explain why a derivative does not exist at a point where a function is continuous, use the limit definition of a derivative and show that the limit does not exist.

#Points of Non-Differentiability on Graphs

If the graph of a function has a point where the tangent to the graph is vertical, then the derivative is undefined at that point.
For example, at $x=0$ on the graph of $f(x) = \sqrt[3]{x}$ , the limit in the limit definition of the derivative becomes unbounded.

#Worked Example

Given the function $f(x) = \sqrt{|x + 6|}$ , determine which of the following statements is true:

(A) $f$ has a vertical asymptote at $x=-6$ .

(B) $f$ is not continuous at $x=-6$ .

(D) $f$ is continuous but not differentiable at $x=-6$ .

Answer:

Consider option (A):

There will be a vertical asymptote if the function becomes unbounded at this point.
Check by substituting in $x=-6$ : $f(-6) = \sqrt{|-6 + 6|} = \sqrt{0} = 0$
The function has a well-defined value of 0 at $x=-6$ , so there is not an asymptote.

Consider option (B):

We have already checked the value of the function at $x=-6$ and it has a value of 0. - The limits from the left and right at $x=-6$ are also equal to 0: $\lim_{{x \to -6^-}} \sqrt{|x+6|} = \sqrt{|-6+6|} = \sqrt{0} = 0$ $\lim_{{x \to -6^+}} \sqrt{|x+6|} = \sqrt{|-6+6|} = \sqrt{0} = 0$
Therefore, the function is continuous at $x=-6$ .

Consider option (C):

The two one-sided limits agree, therefore: $\lim_{{x \to -6}} \sqrt{|x+6|} = 0$

Consider option (D):

By elimination, the answer is D, but we can also check this by inspecting the graph of $f(x)$ .

Common Mistake

You could use your graphing calculator to do this.

- The graph of

f(x) = \sqrt{|x + 6|}

has a cusp at

(-6,0)

, so at this point the function is continuous but not differentiable.

Correct Answer: D

Practice Question

Practice Questions:

Determine whether the following function is differentiable at $x=0$ : $f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases}$ .
Show that the function $f(x) = |x-2|$ is not differentiable at $x=2$ .
Explain why the function $f(x) = \frac{1}{x}$ is not differentiable at $x=0$ .
Determine the points of non-differentiability for the function $f(x) = \sqrt{|x|}$ .

#Glossary

Asymptote: A line that a graph approaches but never touches.
Cusp: A point on a curve where the direction of the curve changes abruptly.
Derivative: A measure of how a function changes as its input changes.
Differentiable: A function is differentiable at a point if its derivative exists at that point.
Limit: The value that a function approaches as the input approaches some value.
Vertical Tangent: A line that touches a curve at a point where the slope of the curve is undefined.

#Summary and Key Takeaways

Differentiability: A function is differentiable at a point if its derivative exists at that point.
Continuity and Differentiability: A function must be continuous at a point to be differentiable at that point, but a continuous function may not always be differentiable.
Non-Differentiability: To show that a function is not differentiable at a point, use the limit definition of the derivative and show that the limit does not exist.
Points of Non-Differentiability: Non-differentiability can occur at cusps, vertical tangents, and points of discontinuity.

Exam Tip

These notes cover the IB curriculum objectives related to the understanding of differentiability and continuity, including the relationship between the two and methods to determine non-differentiability.