Differentiation is the process of finding the rate of change of a function at any point; the derivative measures how a function changes as its input changes.
## Core concept
Differentiation finds the instantaneous rate of change of a dependent variable with respect to an independent variable. If `y = f(x)`, the derivative `dy/dx` or `f'(x)` tells us how quickly `y` changes when `x` changes by an infinitesimal amount.
Key idea: While average rate of change uses two points, differentiation uses the limiting process to find the rate at a single point.
The derivative at a point `x = a` is defined as: ``` f'(a) = lim[h→0] {f(a+h) − f(a)} / h ```
This limit represents the slope of the tangent line to the curve at that point.
## Formula / rule
First Principles (Definition): - `dy/dx = lim[h→0] {f(x+h) − f(x)} / h`
Notation: - `dy/dx` (Leibniz notation) — preferred in calculus - `f'(x)` (Lagrange notation) — compact form - `y'` — shorthand - `Df(x)` — operator notation
Geometric Interpretation: - Derivative = slope of tangent line to the curve at point (x, y) - Non-differentiable points: sharp corners, vertical tangents, discontinuities
## Standard Derivatives (Foundation level)
| Function | Derivative | |----------|-----------| | `f(x) = c` (constant) | `f'(x) = 0` | | `f(x) = x^n` | `f'(x) = nx^(n−1)` | | `f(x) = e^x` | `f'(x) = e^x` | | `f(x) = ln(x)` | `f'(x) = 1/x` | | `f(x) = sin(x)` | `f'(x) = cos(x)` | | `f(x) = cos(x)` | `f'(x) = −sin(x)` |
## Common exam applications
- Finding slope at a point: Calculate derivative, substitute the x-value
- Equation of tangent line: Use point-slope form: `y − y₀ = m(x − x₀)` where `m = f'(x₀)`
- Determining increasing/decreasing intervals: `f'(x) > 0` means increasing; `f'(x) < 0` means decreasing
- Preparing for Maxima/Minima: First derivative test identifies critical points
- Cost/Revenue analysis: Marginal cost = `dC/dx`; marginal revenue = `dR/dx`
## Worked example
Find the derivative of `f(x) = 3x² − 5x + 2` using first principles at `x = 1`.
Using `f'(a) = lim[h→0] {f(a+h) − f(a)} / h`: - `f(1+h) = 3(1+h)² − 5(1+h) + 2 = 3(1+2h+h²) − 5 − 5h + 2 = 3 + 6h + 3h² − 5 − 5h + 2 = h + 3h²` - `f(1) = 3 − 5 + 2 = 0` - `f'(1) = lim[h→0] (h + 3h²)/h = lim[h→0] (1 + 3h) = 1`
## Common mistakes
- Forgetting to apply the chain rule when differentiating composite functions (topic: Differentiation Rules)
- Confusing limits: The limit must exist and be finite for the function to be differentiable
- Sign errors: Watch negative coefficients carefully when applying power rule
- Not checking conditions: Ensure the function is continuous before checking differentiability
- Mixing derivative with integral: Differentiation finds rates; integration finds accumulation
Quick check: Differentiation always reduces the power by 1 in polynomial terms; if power becomes negative, the function is valid only where defined.