Derivative Notation

Direct Answer

The derivative notation captures the instantaneous rate of change of a function. Common forms include dy/dx, f'(x), Df(x), all denoting the slope of the tangent line to the curve y = f(x) at the point x.

Informal Interpretation

When you zoom in on a curve near a point, it begins to look like a straight line. The derivative is the slope of that line. It tells you how steeply f(x) is rising or falling at each instant.

Formal Definition

We define the derivative of f at x = a by the limit

f'(a) = lim_{h → 0} (f(a + h) – f(a))/h.

Equivalently, writing y = f(x),

dy/dx_{|x = a} = lim_{x → a} (f(x) – f(a))/x – a.

Notational Variants

• Leibniz’s notation: dy/dx, d²y/dx²
• Lagrange’s (prime) notation: f'(x), f''(x), f⁽ⁿ⁾(x)
• Euler’s operator: D[f](x), Df(x)
• Newton’s dot notation: ẏ, ÿ

Examples

1. f(x) = x²
   f'(x) = lim_{h → 0} ((x + h)² – x²)/h = 2x.
2. f(x) = sin x
   f'(x) = cos x.
3. y = e^x
   dy/dx = e^x.
4. Quotient rule example:
   d/dx(u/v) = (u'v – uv')/v².

Multivariable and Higher-Order Extensions

• Partial derivatives: ∂f/∂x, ∂²f/∂x∂y
• Gradient vector: ∇f = (∂f/∂x, ∂f/∂y, …)
• Total derivative (Jacobian): Df(ℝⁿ)
• Higher-order: f⁽ⁿ⁾(x), dⁿy/dxⁿ

Common Conventions

• Write f'(a) when emphasizing the point of evaluation.
• Use d/dx[f(x)] to stress the differentiation operator.
• In physics, a dot over a variable (e.g., ẏ) signifies derivative with respect to time.
• Omit the argument when context is clear: f' instead of f'(x).

Next Steps

1. Rules of differentiation (product, chain, implicit differentiation).
2. Geometric and physical interpretations in motion and optimization.
3. Taylor series expansions via higher derivatives.
4. Differential equations, where derivatives define dynamical laws.