2026 Fall Section 3.5 The Chain Rule Part III Related Rates 2
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Overview
Professor Ara Araeipour develops related-rates techniques by differentiating equations implicitly with respect to time, attaching the appropriate rate such as dx/dt or dy/dt, then substituting known values to solve for the unknown. Applications include a cone with height twice its radius, a 20-foot ladder whose base moves at 2 feet per minute, and a deflating sphere losing 8 cubic centimeters per minute.
Key takeaways
- Related-rates differentiation uses time as the independent variable, so every changing quantity contributes its rate, such as d(x²)/dt = 2x dx/dt.
- When a geometric relationship connects variables, substitute that relationship before differentiating if it reduces the equation to the variable whose rate is sought, as h = 2r does for the cone.
- For a 20-foot ladder, x² + y² = 400 links the base and height; when x = 12 feet, y = 16 feet, allowing the height rate to be calculated.
- Choosing a trigonometric equation that uses the provided measurement can simplify an angle-rate problem: cos θ = x/20 directly uses the ladder's base distance.
- Signs communicate direction: a negative dy/dt or dθ/dt indicates decreasing height or angle, while a deflating balloon requires a negative dV/dt.
- Substituting values after differentiation preserves the relationship among changing quantities; the unknown rate can then be solved directly or isolated symbolically first.
Chapters
- For a related-rates problem, treat x and y as functions of time, so differentiating y⁴ produces 4y³ dy/dt and differentiating x² produces 2x dx/dt.
- The example substitutes dx/dt = 10, x = 4, and y = -2 into the differentiated equation to solve for the missing dy/dt.
- Two valid approaches are to substitute known values immediately after differentiation or first isolate the unknown rate algebraically.
- Differentiate 3x + yeˣ = 20 using the product rule on yeˣ, giving 3 dx/dt + eˣ dy/dt + yeˣ dx/dt = 0.
- At x = 0, y = -1, and dy/dt = -8, use e⁰ = 1 to solve for dx/dt.
- The equation's constant right side differentiates to zero, and the unknown rate can be found either by direct substitution or by collecting dx/dt terms first.
- Differentiate an expression containing (5x + y)⁴ with the chain rule, multiplying by 5 dx/dt + dy/dt.
- Substitute x = 2, y = -9, and dx/dt = 3; the inner expression 5x + y evaluates to 1.
- The resulting equation gives dy/dt = -57/4, illustrating how known rates and position values determine an unknown rate.
- A cone's volume is V = (1/3)πr²h; because h = 2r, rewrite it as V = (2/3)πr³ before differentiating.
- Differentiation with respect to time gives dV/dt = 2πr² dr/dt.
- With dV/dt = 54π cubic inches per minute and r = 6 inches, dr/dt = 3/4 inch per minute, so the radius is increasing.
- Represent the ladder setup as a right triangle with base x, height y, and fixed hypotenuse 20 feet, giving x² + y² = 400.
- Differentiate to obtain 2x dx/dt + 2y dy/dt = 0; the base moves away at dx/dt = 2 feet per minute.
- When x = 12 feet, the Pythagorean theorem gives y = 16 feet, and substitution yields dy/dt = -3/2 feet per minute.
- The negative sign means the ladder's height is decreasing at 3/2 feet per minute.
- For the angle θ between the ladder and ground, use cos θ = x/20 because the changing base x is the given side.
- Differentiate with respect to time: -sin θ dθ/dt = (1/20) dx/dt, then use sin θ = y/20.
- At x = 12 feet, y = 16 feet; with dx/dt = 2 feet per minute, dθ/dt = -1/8 radian per minute.
- The negative rate indicates that the angle between the ladder and ground is decreasing.
- For a spherical balloon, V = (4/3)πr³, so differentiating gives dV/dt = 4πr² dr/dt.
- Because the balloon loses 8 cubic centimeters of air per minute, use dV/dt = -8 rather than +8.
- At r = 20 centimeters, solve -8 = 4π(20)² dr/dt to get dr/dt = -1/(200π) centimeter per minute.
- Keep the exact rate unless a decimal approximation is requested; its negative sign means the radius is decreasing.
Transcript excerpt
0:00 is g of x. So, the derivative of y to the power of n is n y to the power of n minus 1 y prime or dy dx. So, this is 3 * 4 12 y cubed dy dt. This is with respect to x, this is with respect to time. t Uh related rates is always about t. So, having said that the last one x squared is 2x dx dt. So, this is something you want to be very careful of. The derivative of x squared is not just 2x. The reason you tag along dx dt because the independent variable is t. So, now we have two choices. We can plug in and that's good enough because that's what they're asking.
0:48 Or we can solve for dy dt. I'm going to show you both ways. First, I'm going to show you the easy way. I'll plug in. So, to plug in, I'm going to replace the dx dt here and here with 10. I'm going to replace the y with -2 and x with 4. Everything is replaced. What is left? dy dt because we don't have it. So, we get 4 * 10 + 12 * -2 cubed dy dt = 2 * 4 * 10. So, this is -8.
1:39 So, over here we get 40. Over here we get -96 dy dt and over here we get 80. Subtract 40. Divide by negative [sighs] 96. And they're both divisible by eight. You get minus five
2:26 Just in case if you don't see they're divisible by eight, just divide by two again and again and you get to the answer. Now, and we're done. This is finished. However, if the question was first find the y dt and then plug in. In other words, they want the y dt as
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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.