2026 Fall Section 3.5 The Chain Rule Part III Related Rates

Overview

Ara Araeipour introduces related rates as a chain-rule application: relate changing quantities with an equation, then differentiate with respect to time. He works through a spherical balloon with dV/dt = 100 cm³/s and diameter 50 cm, finding dr/dt = 1/(25π) cm/s, and expanding lake ripples with dr/dt = 3/2 ft/s and radius 4 ft, finding dA/dt = 12π ft²/s.

Key takeaways

A related-rates solution starts by expressing the quantities in one equation, then differentiates with respect to time so each changing variable contributes its own rate.
For a sphere, V = (4/3)πr³ differentiates to dV/dt = 4πr²(dr/dt), linking volume growth directly to radius growth.
A balloon with dV/dt = 100 cm³/s and diameter 50 cm has dr/dt = 1/(25π) cm/s, or about 0.0127 cm/s.
For a circular ripple, A = πr² differentiates to dA/dt = 2πr(dr/dt); at radius 4 ft and radial growth 3/2 ft/s, dA/dt = 12π ft²/s.
Rate signs convey direction: positive dA/dt or dr/dt indicates the area or radius is increasing.
Exact rates such as 1/(25π) and 12π should be reported before decimal approximations, with units derived from the quantities and time.

Chapters

0:00 Derivative Rules and the Chain Rule Refresher

Reviews the power, product, quotient, and chain rules, alongside derivatives of sine, cosine, tangent, and other trigonometric functions.
States the chain-rule relationship for a composite function and the extended power rule for g(x)^n.
Recaps exponential derivatives, including d/dx(e^x) = e^x and d/dx(b^x) = b^x ln(b).

3:23 Related-Rates Method and the Spherical Balloon Example

Sets out two steps: find an equation relating the quantities, then differentiate both sides with respect to time using the chain rule.
For a sphere, uses V = (4/3)πr³ and D = 2r; the given volume growth is dV/dt = 100 cm³/s.
When the diameter is 50 cm, the radius is 25 cm; substituting into dV/dt = 4πr²(dr/dt) gives dr/dt = 1/(25π) cm/s.
Emphasizes retaining the exact answer and interpreting a positive rate as an increasing radius.

10:00 Circular Ripple Area Rate and a New Time-Derivative Setup

Uses the circle area equation A = πr² for ripples whose radius grows at 3/2 ft/s.
Differentiates with respect to time to obtain dA/dt = 2πr(dr/dt); at r = 4 ft, the area increases at 12π ft²/s, approximately 37.6991 ft²/s.
Explains that a positive dA/dt means the circle’s area is increasing, and that units combine to area per time.
Begins another related-rates exercise asking for dy/dt with dx/dt = 10, x = 4, and y = -2; identifies time t as the independent variable.

Transcript excerpt

0:13 Good morning, class. Here is chapter 3, derivative section 3.5, the chain rule part 3, related rates. It's an application of differentiation, related rates. All right. >> [clears throat] >> So, what did we cover? Derivatives and rates of change, derivative as a function, basic differentiation formulas, product and quotient rule, and the chain rule. We looked at implicit differentiation, and now related rates. Reminding you of various differentiation

1:00 formulas, starting with the definition f prime of x is limit of f of x plus h minus f of x over h as h approaches zero in as Leibniz believed, dy/dx is limit of delta y over delta x over delta x approaches zero, as delta x approaches uh The derivative of x is one, the derivative of c times f of x, c is a number, is a constant, comes out. The derivative of sum is sum of the derivatives. The derivative of differences is the difference of derivatives. Various differentiation formulas, such as the power rule, and as a result of that, the derivative of a constant becomes zero, the derivative of kx becomes k, k is a constant. Trigonometric functions, the derivative of sine is cosine, the

1:47 derivative of cosine is negative sine. The derivative of tan is secant squared, cotangent is negative cosecant squared, secant secant x tan x, and cosec x, the derivative is minus cosec x cotangent x. tangent x. Uh product rule, quotient rule, uh chain rule comes from the composite function y equals f of g, and y prime is f of g prime or f prime of g prime g of x times g prime of x. Now, the extended power rule, d dx of g of x to the power of n is n g of x to the power of n minus one

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Related notes

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Ara Araeipour.