Derivatives
Definition
Differentiation Rules
| Property | Statement |
|---|---|
| Power | \((x^n)'=nx^{{n-1}}\) |
| Product | \((fg)'=f'g+fg'\) |
| Quotient | \((f/g)'=(f'g-fg')/g^2\) |
| Chain | \((f\circ g)'=(f'\circ g)\cdot g'\) |
Examples
Example 1. Differentiate \(f(x)=x^3\sin x\).
Solution. \(f'(x)=3x^2\sin x+x^3\cos x\) by the product rule.
Deep Dive: Derivative
This section builds durable understanding of derivative in calculus through definition-first reasoning, theorem mapping, and error-checking workflows.
Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.
Visual Intuition
Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.
Practice Set
Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.
Target. Your final line should include assumptions, derivation path, and a quick verification.
Practice B. Build an application scenario using derivative and solve it with both symbolic and numeric methods.
Target. Compare outputs and explain any approximation gap.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.