| Term / concept | Definition / short explanation |
|---|---|
| Power rule | If f(x)=a xn with n ∈ ℤ and constant a, then f'(x)=a n xn-1. |
| Sum rule | Derivative of a sum is the sum of derivatives: (f+g)’ = f’ + g’. |
| Constant multiple rule | (c·f)’ = c·f’ for constant c; constants factor out of differentiation. |
| Constant function | If f(x)=k (constant), then f'(x)=0 for all x (horizontal line). |
📌 1. Statement of the rule (power rule)
Let f(x)=axn where a is a constant and n is an integer (n ∈ ℤ).
Then the derivative is:
f'(x) = a(nxn-1)
This single formula covers:
- positive integer powers (x3, x2 etc.),
- zero power: x0=1 so derivative of a·1 is 0,
- negative integer powers (e.g., x−1) — treat n negative and apply same algebraic rule.
Example 1 — simple polynomial
f(x)=4x3 − 5x2 + 7x − 9.
Apply power rule + sum & constant multiple rules termwise:
f'(x)=4·3x2 −5·2x +7·1 − 0 = 12x2 −10x +7.
📌 2. Negative powers & constant functions (explicit)
Negative powers: if f(x)=ax−2, treat n=−2:
f'(x)=a(−2)x−3= −2ax−3.
Constant functions: if f(x)=k then f'(x)=0. Example: derivative of 5 is 0.
🧠 Examiner tip
- Write each term on its own line when differentiating a polynomial — this avoids sign mistakes.
- Always simplify exponent answers and clearly show constants multiplied.
- If asked for f'(x) at a point, compute derivative formula first, then substitute the x-value (show both steps).
📌 3. Short worked examples (with explanation)
Example 2 — negative power
Let g(x)=6x−1. Then g'(x)=6·(−1)x−2= −6x−2 = −6/x2.
Example 3 — derivative at a point
For f(x)=x3−3x2+2 (as earlier), f'(x)=3x2−6x. Evaluate at x=2:
f'(2)=3·4 − 12 = 0. So instantaneous rate of change at x=2 is 0 (stationary).
🌍 Real-world connection
Engineers and scientists use polynomial fits to approximate behaviour; the power rule gives the rate-of-change of these approximations (e.g., velocity as derivative of position polynomial model).
📐 IA spotlight
- Choose a dataset where a polynomial model is plausible (e.g., trajectory data). Fit a polynomial, differentiate the fitted polynomial to estimate instantaneous rates and discuss limits of the model.
- Include discussion of whether integer-power assumption is justified and limitations when using negative powers near x=0.
📌 4. Quick reference & common pitfalls
- Power rule formula: d/dx [a xn] = a n xn-1 (n integer).
- Sum & constant rules: differentiate term-by-term; constants drop out (derivative 0).
- Watch signs: when n negative, n−1 is more negative — simplify carefully (use fraction notation if clearer).
- Undefined at 0: negative powers produce singularities at x=0 — mention domain restrictions if asked.
🔗 Connections (integrated)
- Paper tip (already above): always show derivative formula first, then substitute numbers — examiners expect that order.
- IA & real-world: fit polynomials, differentiate to interpret rates; discuss domain limits when negative powers present.
- TOK: reflect on how mathematical rules (like power rule) are tools that assume differentiability — ask what is lost when models ignore singularities or discontinuities.