How to differentiate from first principles
The limit definition set out step by step, a fully worked cubic example, and why the h in the denominator must cancel before you take the limit.
What the examiner is testing
Whether you understand what a derivative is, rather than which rule to apply. These questions say "from first principles" precisely to stop you writing down the answer, and the marks are almost entirely for the working.
The definition
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$
The method
- Write \( f(x + h) \) by substituting \( x + h \) everywhere \( x \) appears.
- Expand fully. This is where the arithmetic marks are.
- Subtract \( f(x) \). Every term without an \( h \) should now vanish.
- Divide by \( h \) — every remaining term must have an \( h \) to cancel.
- Let \( h \to 0 \), and only now.
Keep the limit notation on every line until step 5.
Worked example
Differentiate \( f(x) = 2x^3 - 5x \) from first principles.
Step 1 — substitute.
$$ f(x + h) = 2(x + h)^3 - 5(x + h) $$
Step 2 — expand. Using \( (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \):
$$ f(x+h) = 2x^3 + 6x^2h + 6xh^2 + 2h^3 - 5x - 5h $$
Step 3 — subtract \( f(x) = 2x^3 - 5x \).
$$ f(x+h) - f(x) = 6x^2h + 6xh^2 + 2h^3 - 5h $$
Notice the \( 2x^3 \) and the \( -5x \) have gone. If any term without an \( h \) survives, the expansion is wrong — this is a free check, so use it.
Step 4 — divide by \( h \). Every term has a factor of \( h \), so it cancels cleanly:
$$ \frac{f(x+h) - f(x)}{h} = 6x^2 + 6xh + 2h^2 - 5 $$
Step 5 — take the limit. Now that no \( h \) sits in a denominator, setting \( h \to 0 \) is legitimate:
$$ f'(x) = \lim_{h \to 0}\left(6x^2 + 6xh + 2h^2 - 5\right) = 6x^2 - 5 $$
Answer: \( f'(x) = 6x^2 - 5 \).
Check with the standard rules: differentiating \( 2x^3 - 5x \) termwise gives \( 6x^2 - 5 \). It matches. You cannot use this as your method, but you can absolutely use it to check.
The three mistakes that lose marks
1. Dropping the limit notation. Write \( \lim_{h \to 0} \) on every line until you actually take the limit. It is worth marks by itself.
2. Expanding the cube wrongly. \( (x + h)^3 \) has four terms. Writing \( x^3 + h^3 \) is the single most common error here.
3. Letting \( h \to 0 \) before cancelling. Doing it at step 3 gives \( 0/0 \). The whole point of the algebra is to reach a form where the limit exists.
30-second recap
Substitute \( x + h \), expand completely, subtract \( f(x) \) and watch the constant terms disappear, cancel the \( h \), then take the limit last. Verify with the power rule afterwards — but show the limit method, because that is what is being marked.