Class 12 Statistics Notes · GSEB
Standard Derivatives
Standard Derivatives — apply the power rule and constant rule with a live power-rule lab. GSEB Class 12 Commerce Statistics notes with worked examples.
Last updated: 24 Sep 2026
Notes
The Two Standard Derivatives
Rule 1 — Power Rule
where n ∈ R and x ∈ R⁺. Bring the exponent down as coefficient, reduce the exponent by 1.
Rule 2 — Constant Rule
A constant does not change, so its rate of change is exactly zero.
Memory hook
Explore: Power Rule Lab
Click through the presets — or dial your own power n — and watch the pattern repeat. After 5–6 examples the rule stops being memorised and starts being obvious.
Function
y = x³
Derivative
dy/dx = 3x²
Cubic growth — slope 3x² means it steepens even faster than x².
Standard Derivative Table
| Function | Derivative | How to see it |
|---|---|---|
| xⁿ | n·xⁿ⁻¹ | bring n down, reduce power by 1 |
| c (constant) | 0 | flat line → slope 0 |
| x | 1 | power rule with n = 1 |
| x² | 2x | power rule with n = 2 |
| x³ | 3x² | power rule with n = 3 |
| 1/x | −1/x² | write as x⁻¹, then power rule |
| √x | 1/(2√x) | write as x^(1/2), then power rule |
| ax + b | a | ax → a, b → 0 |
| axⁿ (a constant) | a·n·xⁿ⁻¹ | constant multiple rule + power rule |
Common trap: y = axⁿ
Why Constant → 0 Makes Sense
The constant rule is not a technicality — it is the maths of things that never move.
Fixed shop rent
₹8,000/month whether you sell 5 cups or 500. Revenue from rent-function R(x) = 8000 → marginal revenue 0.
f(x) = 50
Textbook VSQ: f′(x) = 0. The graph is a horizontal line — slope zero everywhere.
y = aⁿ (a constant)
Another VSQ: dy/dx = 0 — a constant raised to any power is still a constant.
Negative example — the mistake that costs marks
Live Practice Calculator
Question: if f(x) = xⁿ, what is f′(x) evaluated at a chosen point? Set the power n and the point x — the calculator applies f′(x) = n·xⁿ⁻¹ and shows both the derivative value and f(x) for comparison.
Power Rule Evaluator
f′(x) — derivative at x
27
f(x) — original function value
27
Solved Examples
Solved Example
Problem
Solution
f′(x) = 14x − 6
Solved Example
Problem
Solution
dy/dx = 18x² + 7x + 6/5
Solved Example
Problem
Solution
dy/dx = a·n·xⁿ⁻¹
Key Takeaways
Key Takeaways
- ★ Power rule: d/dx (xⁿ) = n·xⁿ⁻¹ — bring the exponent down, reduce it by 1.
- ★ Constant rule: d/dx (c) = 0 — no change means zero rate of change (fixed rent, flat line).
- Coefficients ride along: d/dx (axⁿ) = a·n·xⁿ⁻¹ — the classic MCQ trap.
- Rewrite before differentiating: 1/x = x⁻¹ and √x = x^(1/2), then the same power rule applies.
- These two rules are the raw material for the working rules (sum, product, quotient, chain) in the next topic.
- So what? Marginal cost, marginal revenue and profit optimisation later in the chapter are just these rules wearing business clothes.