Calculus II: Dynamic Systems

Temporal Mechanics

Implicit Differentiation with Respect to Time: The Foundation of Related Rates

Module 01 // Lesson 01

The Dynamic Perspective

Static vs. Dynamic

In standard differentiation, we find how \(y\) changes with respect to \(x\): \(\frac{dy}{dx}\).

In Related Rates, we find how both \(x\) and \(y\) change with respect to time (\(t\)).

Variable \(x\) becomes \(x(t)\)

Variable \(y\) becomes \(y(t)\)

Rates are expressed as \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\)

// The General Chain Rule

\[ \frac{d}{dt}[f(u)] = f'(u) \cdot \frac{du}{dt} \]

Every variable is a function of time. Every differentiation requires a "tail" rate.

A Ripple in a Pond

A pebble is dropped into a still pond, causing a circular ripple to expand. The radius \(r\) of the circle is increasing at a constant rate.

"As the circle grows, is the area growing at the same rate as the radius?"

— Discuss with a partner.

The Differential Mechanism

1

Identify the Core Equation

Start with the geometric or physical relationship (e.g., \(A = \pi r^2\)).

2

Differentiate Both Sides w.r.t. Time (\(t\))

Apply the Power Rule and Chain Rule to every term containing a variable.

\[ \frac{d}{dt}[A] = \frac{d}{dt}[\pi r^2] \implies \frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt} \]

3

Solve for the Unknown Rate

Isolate the rate you are looking for before substituting values.

Guided Practice: Volumetric Expansion

Find the rate of change of the volume of a sphere with respect to time if the radius is changing at rate \(\frac{dr}{dt}\).

Step 1: Formula

\[ V = \frac{4}{3}\pi r^3 \]

The Derivative:

Differentiate: \(\frac{d}{dt}[V]\)

Power Rule: \(\frac{4}{3}\pi \cdot 3r^2\)

Chain Rule: \(\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\)

Danger Zones

Forgetting the Tail

Writing \(2x\) instead of \(2x \frac{dx}{dt}\). In Related Rates, there are no "silent" variables.

Premature Substitution

Plugging in numbers before differentiating. Constants stay, but values at a "specific instant" must wait.

Product Rule Neglect

Differentiating \(xy\) as \(\frac{dx}{dt} \cdot \frac{dy}{dt}\) instead of using \(x \frac{dy}{dt} + y \frac{dx}{dt}\).

Time to Build

Open your "Chain Rule Mastery" worksheets. We will now apply these mechanics to abstract equations before tackling full word problems.

Focus: Precision

Focus: Notation

Focus: Flow

Solving Related Rates Problems in Dynamic Systems

Implicit Differentiation with Respect to Time

Modeling Linear Motion Using Right Triangles

Rates of Change in Expanding Geometric Solids

Calculating Angular Velocity and Trigonometric Rates

Analyzing Fluid Flow in Conical Containers

Temporal Mechanics Slides

Chain Rule Mastery Worksheet

Chain Rule Mastery Answer Key

Linear Motion Slides

Triangle Motion Workshop Worksheet

Triangle Motion Workshop Answer Key

Geometric Solids Slides

Volumetric Growth Activity Worksheet

Volumetric Growth Answer Key

Angular Velocity Slides

Angular Motion Problem Set Worksheet

Angular Motion Answer Key

Conical Flow Slides

Conical Flow Challenge Worksheet

Conical Flow Answer Key

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