Differentiation Rules
Trig Derivatives
| Property | Statement |
|---|---|
| sin | \(\cos x\) |
| cos | \(-\sin x\) |
| tan | \(\sec^2 x\) |
| sec | \(\sec x\tan x\) |
Exponential & Log
| Property | Statement |
|---|---|
| \(e^x\) | \(e^x\) |
| \(a^x\) | \(a^x\ln a\) |
| \(\ln x\) | \(1/x\) |
| \(\log_a x\) | \(1/(x\ln a)\) |
Examples
Example 1. Find \(\frac{{d}}{{dx}}[e^{{x^2}}]\).
Solution. Chain rule: \(2xe^{{x^2}}\).
Deep Dive: Differentiation Rules
This section builds durable understanding of differentiation rules 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 differentiation rules 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.