Differentiation from first principles
The limit definition. The 'why' behind the power rule.
Definition. The derivative of at is
Geometrically, this is the gradient of the tangent to at . The numerator is a secant chord's -change; the denominator is the -change. As the secant approaches the tangent.
Standard exam example: differentiate .
Another example: .
so . Divide by : . Take limit: .
Mark scheme. Always:
- Write down the LIMIT DEFINITION (B1).
- Substitute correctly (M1).
- Expand and simplify (M1).
- Cancel (A1 — the key algebraic step).
- Take the limit (A1).
When to use first principles. ONLY when the question explicitly says 'from first principles'. Otherwise, use the power rule.
- Definition: limit of secant gradient as .
- Step 1: substitute .
- Step 2: expand, cancel .
- Step 3: take the limit (set ).
- Only required when explicitly asked.