Derivatives and Differentiation

What is a Derivative?

The derivative of a function $f(x)$ measures the instantaneous rate of change of the function with respect to $x$. Geometrically, it equals the slope of the tangent line to the curve at a given point.

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

This limit is called the first principles definition of a derivative.

Notation

Several notations are used interchangeably:

| Notation | Meaning | |----------|---------| | $f'(x)$ | Derivative of $f$ with respect to $x$ | | $\dfrac{dy}{dx}$ | Leibniz notation | | $\dot{y}$ | Newton's dot notation (common in physics) | | $D_x f$ | Operator notation |

Fundamental Differentiation Rules

Power Rule

For any real number $n$:

$\frac{d}{dx}[x^n] = n \cdot x^{n-1}$

Examples:

• $\frac{d}{dx}[x^3] = 3x^2$
• $\frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$
• $\frac{d}{dx}[x^{-2}] = -2x^{-3}$

Sum and Difference Rule

$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

Product Rule

$\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$

Mnemonic: "First times derivative of second, plus second times derivative of first"

Example: Differentiate $y = x^2 \sin x$

$\frac{dy}{dx} = 2x \cdot \sin x + x^2 \cdot \cos x = x(2\sin x + x\cos x)$

Quotient Rule

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}$

Example: Differentiate $y = \dfrac{x^2 + 1}{x - 3}$

$\frac{dy}{dx} = \frac{2x(x-3) - (x^2+1)(1)}{(x-3)^2} = \frac{2x^2 - 6x - x^2 - 1}{(x-3)^2} = \frac{x^2 - 6x - 1}{(x-3)^2}$

Chain Rule

For composite functions $y = f(g(x))$:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \quad \text{where } u = g(x)$

Example: Differentiate $y = \sin(3x^2 + 1)$

Let $u = 3x^2 + 1$, so $y = \sin u$:

$\frac{dy}{dx} = \cos u \cdot 6x = 6x\cos(3x^2 + 1)$

Derivatives of Standard Functions

Implementing Numerical Differentiation

def numerical_derivative(f, x: float, h: float = 1e-7) -> float:
    """
    Approximate f'(x) using the central difference formula.
    More accurate than forward difference for small h.
    
    Formula: f'(x) ≈ [f(x+h) - f(x-h)] / (2h)
    """
    return (f(x + h) - f(x - h)) / (2 * h)

import math

# Test with f(x) = sin(x), f'(x) = cos(x)
f = math.sin
x = math.pi / 4  # 45°

approx  = numerical_derivative(f, x)
exact   = math.cos(x)
error   = abs(approx - exact)

print(f"Approximate: {approx:.10f}")
print(f"Exact:       {exact:.10f}")
print(f"Error:       {error:.2e}")  # ~2.3e-14 ✓

Higher-Order Derivatives

The second derivative measures the rate of change of the first derivative (concavity):

$f''(x) = \frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right]$

Concavity test:

• If $f''(x) > 0$: curve is concave up (∪)
• If $f''(x) < 0$: curve is concave down (∩)
• If $f''(x) = 0$: possible inflection point

Applications

Finding Critical Points

At a local maximum or minimum, $f'(x) = 0$ (or undefined).

Algorithm in TypeScript:

interface CriticalPoint {
  x: number;
  type: 'maximum' | 'minimum' | 'inflection' | 'unknown';
}

function classifyCriticalPoint(
  fPrime: (x: number) => number,
  fDoublePrime: (x: number) => number,
  x: number
): CriticalPoint {
  const secondDerivative = fDoublePrime(x);
  
  if (Math.abs(fPrime(x)) > 1e-6) {
    return { x, type: 'unknown' }; // Not actually a critical point
  }

  if (secondDerivative > 0) return { x, type: 'minimum' };
  if (secondDerivative < 0) return { x, type: 'maximum' };
  return { x, type: 'inflection' };
}