Key Idea: This topic finds a derivative from its definition, as the limit of chord gradients. It also covers limits, continuity and second derivatives. A short 'first principles' part is a classic Paper 1 question.
Differentiation from first principles
- Expand — Write and expand f(x + h).
- Subtract and divide — Take away f(x), divide by h.
- Cancel — Cancel the lone h on the bottom.
- Limit — Only now let h → 0.
Limits, continuity & higher derivatives
- A limit is the value f(x) heads towards.
- Continuous: no jumps or holes; the limit equals f(a).
- f″(x) = d²y/dx²: differentiate twice. Nothing is squared.
- Constants vanish at the next derivative.
Cancel h before you let it go to 0, or you get 0/0.
f(x) = x² + 3x from first principles? (2xh + h² + 3h)/h = 2x + h + 3, so f′(x) = 2x + 3.
Why can you not just quote the power rule here? The question asks you to start from the limit.
y = 2x³ + x². Find d³y/dx³. 12.
What does f″ measure? How fast the gradient changes.