First Principles Differentiation

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

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

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

$\lim_{h \to 0}$ 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

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

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).

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

First Principles Differentiation (OCR A Level Maths: Pure)