12.2PureStretch

Finding the Derivative

Differentiation from first principles makes the chord-to-tangent idea precise using a limit. It is the rigorous definition of the derivative — and the reason the power rule you will meet next actually works.

30 min Video by Zeeshan Zamurred Differentiation

Edexcel AS Level Maths: 12.2 Finding the Derivative (First Principles)Watch the full walkthrough before the notes below.

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What you'll be able to do

State the definition of the derivative as a limit
Use first principles to differentiate a simple function
Understand the notation f′(x) and dy/dx
Connect the derivative to the gradient function

The limit definition

Take a point $x$ and a nearby point a tiny step $h$ away. The gradient of the chord is $\frac{f ( x + h ) - f ( x )}{h}$ . Letting $h \to 0$ shrinks the chord to the tangent, giving the derivative.

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

The formal definition of the derivative.

Differentiating from first principles

Substitute the function, expand, simplify the fraction (the $h$ on the bottom cancels), then let $h \to 0$ .

\frac{( x + h ) ^{2} - x ^{2}}{h} = \frac{x ^{2} + 2 x h + h ^{2} - x ^{2}}{h}

2Simplify:

\frac{2 x h + h ^{2}}{h} = 2 x + h

3Let

h \to 0

2 x

Answer

f^{'} (x) = 2 x

Tip — The whole method hinges on the h on the denominator cancelling before you let h → 0.

Notation

The derivative is written $f^{'} (x)$ (function notation) or $\frac{d y}{d x}$ (Leibniz notation) when $y = f (x)$ . Both mean the gradient function — a new function giving the gradient at any $x$ .

y = f (x) \Rightarrow \frac{d y}{d x} = f^{'} (x)

Two notations for the same thing.

Formula recap

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

Derivative from first principles.

\frac{d y}{d x} = f^{'} (x)

Equivalent notations.

f (x) = x^{2} \Rightarrow f^{'} (x) = 2 x

Worked result.

Common mistakes to avoid

Setting h = 0 before cancelling, giving 0/0.

Simplify the fraction first, then take the limit h → 0.

Confusing f′(x) with f(x).

f′(x) is the gradient function, derived from f(x).

Key takeaways

The derivative is the limit of the chord gradient as h → 0.
Method: substitute, expand, cancel the h, then let h → 0.
f′(x) and dy/dx both denote the gradient function.

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