First Principles Differentiation

What is the derivative or gradient function?

For a curve y = f(x) there is an associated function called the derivative or gradient function
The derivative of f(x) is written as f'(x) or $\frac{d y}{d x}$
The derivative is a formula that can be used to find the gradient of y = f(x) at any point, by substituting the x coordinate of the point into the formula
The process of finding the derivative of a function is called differentiation
We differentiate a function to find its derivative

What is differentiation from first principles?

Differentiation from first principles uses the definition of the derivative of a function f(x)

The definition is

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

$\lim_{h \to 0}$ means the 'limit as h tends to zero'
When $h = 0$ , $\frac{f (x + h) - f (x)}{h} = \frac{f (x) - f (x)}{0} = \frac{0}{0}$ which is undefined
- Instead we consider what happens as h gets closer and closer to zero
Differentiation from first principles means using that definition to show what the derivative of a function is

How do I differentiate from first principles?

STEP 1: Identify the function f(x) and substitute this into the first principles formula

e.g. Show, from first principles, that the derivative of 3x² is 6x

f (x) = 3 x^{2}

f' (x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h} = \underset{h \to 0}{l i m} \frac{3 {(x + h)}^{2} - 3 x^{2}}{h^{}}

STEP 2: Expand f(x+h) in the numerator

f' (x) = \underset{h \to 0}{l i m} \frac{3 (x^{2} + 2 h x + h^{2}) - 3 x^{2}}{h}

f' (x) = \underset{h \to 0}{l i m} \frac{3 x^{2} + 6 h x + 3 h^{2} - 3 x^{2}}{h}

STEP 3: Simplify the numerator, factorise and cancel h with the denominator

f' (x) = \underset{h \to 0}{l i m} \frac{h (6 x + 3 h)}{h}

STEP 4: Evaluate the remaining expression as h tends to zero

f' (x) = \underset{h \to 0}{l i m} (6 x + 3 h) = 6 x

A s h \to 0, (6 x + 3 h) \to (6 x + 0) \to 6 x

∴

The derivative of

3 x^{2}

6 x

Examiner Tip

Most of the time you will not use first principles to find the derivative of a function (there are much quicker ways!). However, you can be asked on the exam to demonstrate differentiation from first principles.
Make sure you can use first principles differentiation to find the derivatives of kx, kx² and kx³ (where k is a constant).

Worked example

1st Princ Diff Example, A Level & AS Maths: Pure revision notes

First Principles Differentiation (AQA AS Maths: Pure)

Revision Note

What is the derivative or gradient function?

What is differentiation from first principles?

How do I differentiate from first principles?

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1. Proof

1.1 Proof

1.1.1 Language of Proof

1.1.2 Proof by Deduction

1.1.3 Proof by Exhaustion

1.1.4 Disproof by Counter Example

2. Algebra & Functions

2.1 Laws of Indices & Surds

2.1.1 Laws of Indices

2.1.2 Manipulating Surds

2.1.3 Surds - Rationalising the Denominator

2.2 Quadratics

2.2.1 Quadratic Graphs

2.2.2 Discriminants

2.2.3 Completing the square

2.2.4 Solving Quadratic Equations

2.2.5 Further Solving Quadratic Equations (Hidden Quadratics)

2.3 Simultaneous Equations

2.3.1 Linear Simultaneous Equations - Elimination

2.3.2 Linear Simultaneous Equations - Substitution

2.3.3 Quadratic Simultaneous Equations

2.4 Inequalities

2.4.1 Linear Inequalities

2.4.2 Quadratic Inequalities

2.4.3 Inequalities on Graphs

2.5 Polynomials

2.5.1 Expanding Brackets

2.5.2 Polynomial Division

2.5.3 Factor Theorem

2.5.4 Factorisation

2.6 Graphs of Functions

2.6.1 Sketching Polynomials

2.6.2 Reciprocal Graphs - Sketching

2.6.3 Solving Equations Graphically

2.6.4 Proportional Relationships

2.7 Transformations of Functions

2.7.1 Translations

2.7.2 Stretches

2.7.3 Reflections

2.8 Modelling with Functions

2.8.1 Modelling with Functions

3. Coordinate Geometry

3.1 Equation of a Straight Line

3.1.1 Basic Coordinate Geometry

3.1.2 Parallel & Perpendicular Gradients

3.1.3 Equation of a Straight Line

3.1.4 Modelling with Straight Lines

3.2 Circles

3.2.1 Equation of a Circle

3.2.2 Finding the Centre & Radius

3.2.3 Bisection of Chords

3.2.4 Angle in a Semicircle

3.2.5 Radius & Tangent

4. Sequences and Series

4.1 Binomial Expansion

4.1.1 Binomial Expansion

5. Trigonometry

5.1 Basic Trigonometry

5.1.1 Trigonometry - Definitions

5.1.2 Right-Angled Triangles

5.1.3 Non-Right-Angled Triangles

5.2 Trigonometric Functions

5.2.1 Graphs of Trigonometric Functions

5.2.2 Transformations of Trigonometric Functions

5.3 Trigonometric Equations

5.3.1 Trigonometry - Simple Identities

5.3.2 Linear Trigonometric Equations

5.3.3 Quadratic Trigonometric Equations

5.3.4 Strategy for Trigonometric Equations

6. Exponentials & Logarithms

6.1 Exponential & Logarithms

6.1.1 Exponential Functions