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

Implicit Differentiation with Respect to Time

Temporal Mechanics Slides

Chain Rule Mastery Worksheet

Chain Rule Mastery Answer Key

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