Physical Meaning of the Derivative

Derivatives in physics express instantaneous rates of change. They tie motion, force, fields, thermodynamics, and even economic growth to a single mathematical idea.

Instantaneous rate of change

For a quantity $y$ depending on $x$,

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

Kinematics

• Velocity: displacement rate
  $v(t) = \frac{ds}{dt} = \lim_{\Delta t \to 0} \frac{s(t + \Delta t) - s(t)}{\Delta t}$
• Acceleration: velocity rate
  $a(t) = \frac{dv}{dt} = \frac{d^2 s}{dt^2}$
• Jerk: acceleration rate
  $j(t) = \frac{da}{dt} = \frac{d^3 s}{dt^3}$

Mechanics

• Power: $P(t) = \frac{dW}{dt}$
• Force (momentum rate): $\vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt}$
  For constant mass: $\vec{F} = m\vec{a}$.
• Angular motion: $\omega(t) = \frac{d\theta}{dt}, \quad \alpha(t) = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$

Electromagnetism

• Current: $i(t) = \frac{dq}{dt}$
• Induced EMF: $\varepsilon = -\frac{d\Phi}{dt}$

• Inductor voltage: $V_L = L\frac{di}{dt}$
• Capacitor current: $i_C = C\frac{dV}{dt}$

Thermodynamics

• Heat capacity: $C = \frac{dU}{dT}$
• Entropy change: $dS = \frac{dQ}{T}$
• Temperature gradient: $\nabla T = \frac{dT}{dx}$

Fluid mechanics

• Density: $\rho = \frac{dm}{dV}$
• Pressure: $P = \frac{dF}{dA}$
• Flow rate: $Q = \frac{dV}{dt}$

Optics

• Refractive index gradient: $\nabla n = \frac{dn}{dx}$
• Light intensity: $I = \frac{dE}{dt}$

Economics

• Marginal cost: $MC = \frac{dC}{dQ}$
• Marginal revenue: $MR = \frac{dR}{dQ}$
• Price elasticity: $E = \frac{dQ}{dP} \cdot \frac{P}{Q}$

Biology

• Growth rate: $r = \frac{dN}{dt}$
• Metabolic rate: $M = \frac{dE}{dt}$

Exercises

练习 1

For $s(t) = 2t^2 + 3t + 1$, find (1) $v(t)$ (2) $a(t)$ (3) $v(2)$ and $a(2)$.

练习 2

For a capacitor with $V(t) = 10\sin(2t)$ and $C = 2\ \text{F}$, find $i(t)$.

练习 3

If $m(V) = 2V^2 + 3V$, find the density $\rho(V)$.

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Summary

本文出现的符号

中英对照

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