derive

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# Mathematical Derivation Skill

## Purpose

Produce rigorous, step-by-step mathematical derivations from premises to conclusion. Every step is justified, every variable is defined, and the output is formatted for direct inclusion in academic reports.

## Supported Domains

- **Linear algebra**: Matrix operations, eigenvalue problems, decompositions, trace/determinant identities
- **Calculus**: Differentiation, integration, Taylor/Maclaurin series, limits, multivariable calculus
- **Probability and statistics**: Bayes' theorem, expectations, distributions, maximum likelihood, Bayesian inference
- **Optimisation**: Gradient descent convergence, Lagrange multipliers, KKT conditions, convexity proofs
- **Differential equations**: ODEs, PDEs, separation of variables, Green's functions, weak formulations
- **Numerical methods**: Discretisation error analysis, stability proofs, convergence rates
- **Machine learning**: Backpropagation, loss function gradients, kernel tricks, information-theoretic bounds

## Workflow

### Step 1: Clarify the Derivation

Before starting, establish:

1. **Starting point**: What are the premises, axioms, or given equations?
2. **Target**: What expression, identity, or result needs to be derived?
3. **Level of rigour**: Sketch (key steps only), standard (all steps, informal justification), or formal (proof-level, every inference rule named)
4. **Notation conventions**: Does the user follow specific notation? (e.g., bold for vectors, hat for estimates, bar for means)
5. **Context**: Is this for a report, homework, or personal understanding? This affects the level of explanation.

### Step 2: Define All Variables

Before the first equation, produce a notation table:

```
Let:
- A ∈ ℝ^{m×n}    : the coefficient matrix
- x ∈ ℝ^n         : the solution vector
- b ∈ ℝ^m         : the right-hand side vector
- ‖·‖_2           : the Euclidean (L2) norm
- κ(A)            : the condition number of A, defined as ‖A‖‖A⁻¹‖
```

Every variable used in the derivation must appear in this table. No symbol should appear without definition.

### Step 3: Perform the Derivation

Structure each step as:

```
Step k: [Brief description of what this step does]

[Equation from previous step]

    = [New expression]          (by [justification])

    = [Next expression]         (by [justification])

[1-3 sentences explaining WHY this step was taken and WHAT it accomplishes]
```

#### Justification Types (always specify one):

| Justification | When to Use |
|---------------|-------------|
| by definition | Substituting a definition |
| by [Named Theorem] | Applying a known theorem (state it) |
| by [algebraic operation] | Factoring, expanding, combining like terms |
| by substitution of (k) | Substituting result from step k |
| by linearity of [operator] | Using linearity of expectation, integral, etc. |
| by chain rule | Composite function differentiation |
| by product rule | Product differentiation |
| by integration by parts | The standard IBP formula |
| by Cauchy-Schwarz | Applying the inequality |
| by triangle inequality | Applying |a+b| ≤ |a|+|b| |
| by assumption | Something stated in the premises |
| by symmetry of A | Using A = A^T |
| by positive definiteness | Using x^T A x > 0 for x ≠ 0 |
| rearranging | Simple algebraic manipulation |

### Step 4: Handle Branches and Cases

When a derivation requires case analysis:

```
We consider two cases:

Case 1: [Condition]
  [Derivation for this case]
  ∴ [Result for case 1]

Case 2: [Condition]
  [Derivation for this case]
  ∴ [Result for case 2]

Combining both cases: [Final result]
```

### Step 5: Verify the Result

After completing the derivation, perform at least one verification:

- **Dimensional analysis**: Do the units/dimensions match on both sides?
- **Special case check**: Substitute a simple known case (e.g., n=1, A=I) and verify
- **Boundary check**: Does the result behave correctly at extremes (x→0, x→∞)?
- **Numerical check**: Plug in specific numbers and verify both sides agree
- **Symmetry check**: If the result should be symmetric in certain variables, verify it is

State which verification was performed and show the check.

### Step 6: Output Format

Provide the derivation in two formats:

#### A. Annotated Markdown (for understanding)
Each step with full English explanation, motivation, and commentary. Use this for teaching and personal notes.

#### B. LaTeX (for reports)
Clean, publication-ready LaTeX using proper environments:

```latex

From rafeeqinea/claude-academic-toolkit, MIT.

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