Limit definition of the derivative

IB Mathematics: Applications and Interpretation HL· 10 min read

1. Geometric Motivation & Formal Definition★★☆☆☆⏱ 15 min

To find the slope of a tangent line at a point $x = a$ , we start with the slope of a secant line between $x = a$ and $x = a + h$ , then take the limit as $h$ approaches 0, making the distance between the two points infinitely small.

📘 Definition

Derivative at a point

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

The derivative $f^{'} (a)$ is the slope of the tangent line to $y = f (x)$ at $x = a$ , equal to the instantaneous rate of change of $f (x)$ at $x = a$ .

📐 Worked Example

Use the limit definition to find $f^{'} (1)$ for $f (x) = 3 x^{2}$ .

1
Step 1: Calculate $f (1)$ :
2
$f(1) = 3(1)^2 = 3$
3
Step 2: Calculate $f (1 + h)$ :
4
$f(1+h) = 3(1+h)^2 = 3(1 + 2h + h^2) = 3 + 6h + 3h^2$
5
Step 3: Simplify the difference quotient:
6
$\frac{f(1+h) - f(1)}{h} = \frac{(3 + 6h + 3h^2) - 3}{h} = \frac{6h + 3h^2}{h} = 6 + 3h, \quad h \neq 0$
7
Step 4: Evaluate the limit as $h \to 0$ :
8
$f'(1) = \lim_{h \to 0} (6 + 3h) = 6$

2. Differentiating from First Principles★★★☆☆⏱ 20 min

We can also find a general derivative function $f^{'} (x)$ that gives the derivative at any value of $x$ , rather than just a single point $x = a$ . Calculating the derivative directly from the limit definition is called differentiating from first principles.

📘 Definition

General Derivative Function

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The derivative function returns the slope of the tangent at any $x$ where the limit exists.

📐 Worked Example

Differentiate $f (x) = x^{2} + 4 x$ from first principles.

1
Step 1: Expand $f (x + h)$ :
2
$f(x+h) = (x+h)^2 + 4(x+h) = x^2 + 2xh + h^2 + 4x + 4h$
3
Step 2: Subtract $f (x)$ to find the change in $f$ :
4
$f(x+h) - f(x) = (x^2 + 2xh + h^2 + 4x + 4h) - (x^2 + 4x) = 2xh + h^2 + 4h$
5
Step 3: Divide by $h$ to get the difference quotient:
6
$\frac{f(x+h) - f(x)}{h} = 2x + h + 4$
7
Step 4: Take the limit as $h \to 0$ :
8
$f'(x) = \lim_{h \to 0} (2x + h + 4) = 2x + 4$

✓ Quick check

Check your understanding of the definition structure

What is the correct difference quotient for $f^{'} (a)$ ?
- $\frac{f ( a + h ) - f ( a )}{h}$
- $\frac{f ( a + h )}{h}$
- $\frac{f ( a + h ) - f ( a )}{a}$
- $f (a + h) - f (a)$

3. Non-Differentiability and the Limit Definition★★★★☆⏱ 15 min

The derivative only exists at a point if the two-sided limit of the difference quotient exists. Common cases of non-differentiability include discontinuities, corner points, and vertical tangents.

📐 Worked Example

Show $f (x) = ∣ x - 3∣$ is not differentiable at $x = 3$ using the limit definition.

1
Step 1: Split the absolute value into piecewise: $∣ x - 3∣ = x - 3$ for $x > 3$ , $∣ x - 3∣ = 3 - x$ for $x < 3$ .
2
Step 2: Calculate the left-hand limit ( $h < 0$ , so $3 + h < 3$ ):
3
$\lim_{h \to 0^-} \frac{f(3+h) - f(3)}{h} = \lim_{h \to 0^-} \frac{|h|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$
4
Step 3: Calculate the right-hand limit ( $h > 0$ , so $3 + h > 3$ ):
5
$\lim_{h \to 0^+} \frac{|h|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1$
6
Step 4: Conclusion: Since left-hand limit $\neq =$ right-hand limit, the two-sided limit does not exist, so $f^{'} (3)$ does not exist.

4. Common Pitfalls

Wrong move:

Forgetting to cancel $h$ from the numerator before substituting $h = 0$ , leading to division by zero.

Why:

The difference quotient is undefined at $h = 0$ , so you must simplify first before evaluating the limit.

Correct move:

Always factor $h$ out of the numerator, cancel it with the denominator, then substitute $h = 0$ to find the limit.

Wrong move:

Omitting the limit notation when writing the derivative expression.

Why:

The derivative is the limit of the difference quotient, not the difference quotient itself. Omitting the limit changes the meaning entirely.

Correct move:

Always include $lim_{h \to 0}$ in any expression for the derivative calculated from the definition.

Wrong move:

Assuming continuous functions are always differentiable.

Why:

Continuity is a necessary but not sufficient condition for differentiability. Corners and cusps are continuous but not differentiable.

Correct move:

Always check that the two-sided limit of the difference quotient exists to confirm differentiability.

Wrong move:

Making an error expanding $(x + h)^{n}$ for $n \geq 3$ , leading to an incorrect final derivative.

Why:

Misexpanding the binomial is a common algebraic mistake when working from first principles.

Correct move:

Double-check binomial expansion, or use the alternative definition with factoring to simplify the expression.

5. Quick Reference Cheatsheet

Concept	Formal Expression	Use Case
Derivative at $x = a$	$f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$	Find slope at one point
Alternative derivative at $x = a$	$f^{'} (a) = lim_{x \to a} \frac{f ( x ) - f ( a )}{x - a}$	Easy polynomial factoring
General derivative function	$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$	Find derivative for all $x$
Non-differentiability check	$lim_{h \to 0^{-}} \neq = lim_{h \to 0^{+}}$	Confirm corners/cusps

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · Paper 1
First principles derivative calculation
2023 · Paper 1
Check non-differentiability at a point

What's Next

The limit definition of the derivative is the foundational concept for all of calculus. Now that you understand where derivatives come from, you can move on to learn the shortcut differentiation rules that allow you to compute derivatives quickly for exams. These rules, built on the limit definition, will be used for all applications of calculus including optimization, integration, and modeling real-world phenomena in IB AI HL. You will also use the definition to understand when derivatives do not exist, which is important for graphing functions and solving optimization problems.