First principles differentiation

The idea

The derivative is the limit of the gradient of a chord: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ . Simplify the fraction first, then let $h$ tend to $0$ .

Write $f(x + h) - f(x)$ and expand.
Divide every term by $h$ .
Let $h \to 0$ : terms with $h$ in them vanish.

Watch out Cancel the $h$ before letting $h \to 0$ , otherwise you get $\frac{0}{0}$ .

See it worked

Worked example

Differentiate $f(x) = x^2$ from first principles.

$f(x + h) - f(x) = (x + h)^2 - x^2 = 2xh + h^2$ .
$\frac{2xh + h^2}{h} = 2x + h$ .
As $h \to 0$ , $2x + h \to 2x$ , so $f'(x) = 2x$ .

Answer $f'(x) = 2x$

Your turn

Question 1Non-calculator3 marks

Use first principles to find $f'(x)$ for $f(x) = x^2 + 3x$ .

Question 2Non-calculator3 marks

Find $\lim_{h \to 0} \frac{(3 + h)^2 - 9}{h}$ .

Question 3Non-calculator3 marks

For $f(x) = x^3$ , the chord from $x = 2$ to $x = 2 + h$ has gradient $12 + 6h + h^2$ . Find $f'(2)$ .

In past papers

17 official questions in 17 papers since 2018. The latest:

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