Differentiation

The derivative measures instantaneous rate of change and slope. Learn the limit definition, geometric and physical meaning, standard derivatives, product, quotient and chain rules, higher derivatives, implicit, parametric, logarithmic and inverse-function differentiation, and an introduction to partial derivatives.

Velocity is the derivative of position; current is the derivative of charge; strain rate, heat flux and power are all derivatives. Differentiation tells engineers how fast things change and how sensitive outputs are to inputs.

An analogy

The derivative is a speedometer: it tells you how fast the output is changing right now, not the average over the whole trip.

Meaning

Geometrically, f′(a) is the slope of the tangent line at x = a. Physically, it is a rate: m/s, A = C/s, W = J/s. Units of dy/dx are units of y divided by units of x.

Rules

Linearity (sum and constant multiple), product rule, quotient rule and chain rule handle almost every elementary function. The chain rule is the most used — and most forgotten — rule.

Special techniques

Implicit: differentiate both sides of F(x, y) = 0 treating y as y(x). Logarithmic: take ln first for products of powers or xˣ. Inverse functions: (f⁻¹)′ = 1/f′. Parametric: divide the t-derivatives.

Higher and partial derivatives

Second derivatives give acceleration and curvature; beam bending uses up to the fourth derivative (EI y⁗ = w). For f(x, y), ∂f/∂x varies x holding y fixed — the start of multivariable calculus.

Formulas

• Definition: f′(x) = lim_{h→0} [f(x + h) − f(x)]/h
• Standard derivatives: d(xⁿ)/dx = nxⁿ⁻¹, d(sin x) = cos x, d(cos x) = −sin x, d(eˣ) = eˣ, d(ln x) = 1/x
• Product & quotient: (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²
• Chain rule: d/dx f(g(x)) = f′(g(x)) · g′(x)
• Parametric: dy/dx = (dy/dt)/(dx/dt)

Worked example: Chain and product rules

Given: y = x² sin(3x)

Find: dy/dx.

• u = x², v = sin 3x
• u′ = 2x, v′ = 3cos 3x (chain rule)
• y′ = 2x sin 3x + 3x² cos 3x

Answer: dy/dx = 2x sin 3x + 3x² cos 3x

Worked example: Implicit differentiation

Given: x² + y² = 25

Find: dy/dx at (3, 4).

• 2x + 2y y′ = 0
• y′ = −x/y = −3/4

Answer: −3/4 (tangent to the circle)

Common mistakes

• Wrong: d/dx sin(3x) = cos(3x). — Right: Chain rule: 3cos(3x).
• Wrong: (uv)′ = u′v′. — Right: (uv)′ = u′v + uv′.

Applications

• Kinematics: v = ds/dt, a = dv/dt.
• Circuits: i = C dv/dt, v = L di/dt.
• Sensitivity: How outputs respond to parameter changes.

Key takeaways

• f′ = lim [f(x + h) − f(x)]/h.
• Product, quotient, chain rules.
• Implicit, parametric, logarithmic techniques.
• Derivatives are rates with units.

Going further

• Automatic differentiation: Exact derivatives in software (ML, optimisation).
• Fractional derivatives: Viscoelastic materials and control.
• Differentials: dy = f′(x) dx for error estimates.