4.4 Intro to Related Rates

Step-by-step you can always follow this checklist for Topic 4.4 related-rates problems: 1. Read context and assign variables with units (e.g., x(t), y(t), r(t)). Identify which rate is given and which you need (dx/dt, dy/dt). 2. Write an equation relating the variables (geometry, similar triangles, Pythagorean, volume formulas). 3. Differentiate both sides wrt time t using the chain rule (and product/quotient/implicit as needed). That gives an equation in rates (e.g., 2x dx/dt + 2y dy/dt = 0). 4. Substitute the known variable values at the instant and the known rates (use correct units). 5. Solve algebraically for the unknown rate and state it with units and sign (positive = increasing). 6. Check reasonableness (limits, sign, magnitude). Key AP tips: always label dx/dt, dy/dt, use implicit differentiation (chain rule is the foundation per CED CHA-3.D), and show work—this is exactly what AP free-response expects. For extra examples and practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2), the Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4), and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Related-rates problems use the chain rule: you differentiate an equation relating several changing quantities with respect to the same independent variable (usually time t), then solve for the desired rate (like dy/dt). Two useful templates: - If F(x, y, …) = 0, implicitly differentiate: F_x dx/dt + F_y dy/dt + … = 0 (sum of partials times the corresponding rates). - If z = f(x,y), use the multivariable chain rule: dz/dt = f_x(x,y)·dx/dt + f_y(x,y)·dy/dt. Concrete example (CED keywords: chain rule, implicit differentiation): for a cylinder V = πr^2h, dV/dt = 2πr h·dr/dt + πr^2·dh/dt so you plug known r, h, and one or two known rates to find the unknown. This is exactly what Topic 4.4 tests (CHA-3.D in the CED). For more worked examples and practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and the unit page (https://library.fiveable.me/ap-calculus/unit-4).

Use the chain rule any time you’re differentiating a quantity that depends on a variable which itself depends on time—in related rates you always differentiate every variable with respect to t (that’s CHA-3.D). Use the product (or quotient) rule when the equation you’re differentiating has a product (or quotient) of two or more time-varying quantities. Quick recipe: - Step 1: Write the relation (e.g., x^2 + y^2 = 25 or V = πr^2h). - Step 2: Differentiate both sides with respect to t. For each variable use the chain rule: d/dt[f(x)] = f′(x)·dx/dt. - Step 3: If a term is a product (like r^2·h), apply the product rule, remembering to use chain rule inside: d/dt(r^2·h) = (2r dr/dt)·h + r^2·dh/dt. Examples: - Implicit (circle): d/dt(x^2 + y^2) = 2x dx/dt + 2y dy/dt (chain rule). - Cylinder V=πr^2h: dV/dt = π[2r dr/dt · h + r^2 dh/dt] (product + chain). This is exactly what the CED expects for related rates (CHA-3.D and CHA-3.D.2). For more worked examples and practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

You use the chain rule / implicit differentiation: write an equation relating x and y, differentiate every term with respect to time t (treat dx/dt and dy/dt as derivatives), then solve for dy/dt by plugging the known dx/dt and the point values. Quick example (common on the AP): x^2 + y^2 = 25. Differentiate w.r.t. t: 2x(dx/dt) + 2y(dy/dt) = 0. Solve for dy/dt: dy/dt = −[x(dx/dt)] / y. So if dx/dt = 3 cm/s and the point is (4, 3), dy/dt = −(4·3)/3 = −4 cm/s. Steps to follow on any problem: 1) Write the geometric/physical relation (Pythagorean, similar triangles, volume formula...). 2) Differentiate implicitly w.r.t. t using chain/product/quotient rules as needed (CHA-3.D). 3) Plug in x, y, dx/dt values at the instant given and solve for dy/dt. This is exactly what the AP CED expects for related rates (Topic 4.4). For more worked examples and practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Look for any quantity in the story that depends on another quantity—those are your related variables. Steps you can use every time: - Identify what’s changing (rates asked for: e.g., dV/dt, dx/dt). Write variables for those quantities with units. - Find a connecting equation from geometry/physics (Pythagorean for ladders, V = (1/3)πr^2h for cones, similar triangles for shadow problems, area = πr^2 for balloons). That equation shows how the variables relate. - Decide which variables are independent/dependent in context (time t is the common independent variable in related rates: use chain rule and write dx/dt, dr/dt, dh/dt). - Implicitly differentiate the connecting equation with respect to t, using product/quotient/chain rules as needed (CED CHA-3.D.1 & CHA-3.D.2). - Plug known values and solve for the requested rate. If you’re unsure which equation to use, scan for geometric relations or conservation laws in the text. For examples and guided practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Implicit differentiation is a differentiation technique: when an equation defines y implicitly (like x^2 + y^2 = 25), you differentiate both sides with respect to x and use dy/dx for every y-term (and product/quotient rules as needed). Related rates is an application: you have several quantities that change with time, so you differentiate an equation that ties them together with respect to the same independent variable (usually t). In practice, related rates uses implicit differentiation + the chain rule (replace dy/dt for y′, dx/dt for x′, etc.) to get equations relating rates (e.g., dx/dt, dV/dt). Key AP points: Topic 4.4 expects you to differentiate with respect to time, use chain/product/quotient rules, and interpret units (CHA-3.D). Common contexts: ladder, balloon, conical tank, shadow problems. For a focused review see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Pick variables, write relation, differentiate, solve. Example: A spherical balloon’s volume V = (4/3)πr^3. Let r = r(t), V = V(t). If the volume is increasing at dV/dt = 8 cm^3/s when r = 2 cm, find dr/dt. 1. Differentiate both sides with respect to time t (use chain rule / implicit differentiation): dV/dt = 4πr^2 · dr/dt. 2. Solve for dr/dt: dr/dt = (dV/dt) / (4πr^2) = 8 / (4π·2^2) = 8 / (16π) = 1/(2π) cm/s. So the radius is increasing at 1/(2π) cm/s at that instant. Always state units and what each rate means (dV/dt in cm^3/s, dr/dt in cm/s). On the AP exam show: variable definitions, the equation, differentiation step (chain rule), substitution, and units—that’s aligned with CHA-3.D (CED). For more worked examples and practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and try problems at Fiveable’s practice page (https://library.fiveable.me/practice/ap-calculus).

You use the chain rule in related-rates problems because you’re differentiating quantities that depend on the same hidden independent variable (usually time). If an equation ties two or more changing quantities together (for example, V = πr^2h or x^2 + y^2 = 25), each variable is really a function of t. The chain rule tells you how to take d/dt of those expressions: differentiate with respect to the variable in the equation, then multiply by that variable’s rate (dx/dt, dy/dt, etc.). That’s exactly CHA-3.D from the CED: “The chain rule is the basis for differentiating variables in a related rates problem with respect to the same independent variable.” Often you combine chain rule with implicit differentiation and other rules (product/quotient) to get an equation involving the desired rates, then plug in known values. For more examples and step-by-step practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Before you differentiate, build one clear equation that relates the changing quantities (use one equation only). Steps: 1. Identify all variables and what rates you’re given/asked for (use dx/dt notation and include units). 2. Draw a diagram and label the variables—geometry often helps (Pythagorean, similar triangles, volume formulas, area, etc.). 3. Write one relationship connecting the variables (e.g., x^2 + y^2 = 25 for a ladder, V = (1/3)πr^2h for a cone, or similar-triangle ratios for shadows). 4. If needed, eliminate extra variables using a second algebraic relation so the main equation contains only the variables you’ll differentiate. 5. Check units and which variable is implicitly a function of time (t). Only then differentiate implicitly with respect to t, using the chain rule—plus product or quotient rules if they appear—and substitute the known values to solve for the requested rate. This follows CED ideas (CHA-3.D: chain rule, implicit differentiation; use product/quotient as needed). For examples and guided practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and try problems from Fiveable’s practice set (https://library.fiveable.me/practice/ap-calculus).

Short answer: almost always differentiate first, then substitute the known numerical values. Why: related-rates problems use implicit differentiation with respect to the same independent variable (usually t). Treat every letter as a function of t, apply the chain rule (and product/quotient rules if needed) to get an equation relating the rates (dx/dt, dy/dt, etc.). Only after you have that equation do you plug in the specific values of the variables and any known rates to solve for the unknown rate. Quick steps: 1) Write the equation linking the quantities (e.g., x^2 + y^2 = r^2). 2) Differentiate implicitly with respect to t: 2x dx/dt + 2y dy/dt = 2r dr/dt. 3) Substitute the given numbers (x, y, known rates) and solve for the unknown rate. Exception: if a letter really is a fixed constant (not changing with t), you can substitute it beforehand—but check the problem carefully. This approach matches CED CHA-3.D (chain rule, implicit differentiation). For more examples and practice, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4). For lots of practice problems, go to (https://library.fiveable.me/practice/ap-calculus).

When a problem asks for a "rate of change with respect to time," it simply means take the derivative of a quantity using time (usually t) as the independent variable—e.g., dx/dt, dV/dt, dr/dt. Practically, that derivative tells you how fast that quantity is changing per one unit of time and includes units (meters/second, liters/minute). The sign matters: positive means increasing, negative means decreasing. In related-rates problems (Topic 4.4 on the CED) you usually have two or more variables related by an equation (Pythagorean theorem, volume formula, similar triangles, etc.). Differentiate that equation implicitly with respect to t (use chain rule, product/quotient rules as needed) to get an equation in the time-derivatives (dx/dt, dy/dt, dV/dt). Then plug in known values to solve for the unknown rate. For review and worked examples, check the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and do lots of practice problems (https://library.fiveable.me/practice/ap-calculus)—AP free-response often tests this skill, so practice implicit differentiation and interpreting units.

Start by labeling: let x(t) = distance from wall to ladder base, y(t) = ladder top height, and L = ladder length (constant). Use the Pythagorean relation x^2 + y^2 = L^2. Differentiate implicitly with respect to time t (chain rule / implicit differentiation): 2x dx/dt + 2y dy/dt = 0. Solve for the unknown rate (usually dy/dt): dy/dt = −(x/y) · dx/dt. Steps to solve any ladder problem 1. Draw and label x, y, L; write the geometric equation. 2. Differentiate both sides w.r.t. t using chain rule and other rules if needed (CHA-3.D). 3. Plug in the given x, y (or compute y from L and x) and the known rate (dx/dt or dy/dt). 4. Compute sign and units—a negative dy/dt means the top is sliding down. Quick numeric example: L = 10 ft, base moves away at dx/dt = 3 ft/s when x = 6 ft. Then y = 8 ft, so dy/dt = −(6/8)(3) = −2.25 ft/s (top moving down at 2.25 ft/s). For more worked examples and AP-aligned practice on related rates, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Short answer: you don’t have to literally write d/dt every time, but you must show you’re differentiating with respect to the same independent variable (usually time). AP CED Topic 4.4 expects you to use the chain rule/implicit differentiation to turn relationships among variables into relationships among their rates (CHA-3.D.1/CHA-3.D.2). Why d/dt is helpful: writing dy/dt, dx/dt (or dV/dt) makes it explicit that x, y, V are functions of t and keeps units clear. Using y′ or x′ is fine too—as long as you state what the prime means (e.g., x′ = dx/dt). The key error exam graders watch for is treating dx and dy as independent or forgetting the common independent variable. Example: if x^2 + y^2 = 25 and x = x(t), y = y(t), differentiate with respect to t: 2x dx/dt + 2y dy/dt = 0 (showing you used d/dt). That’s the AP-style approach. For practice and more worked examples, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Start by reading the scenario and asking: what physical quantity do they want as a rate (e.g., how fast is the radius changing—dr/dt—or how fast is the volume changing—dV/dt)? Steps that make that clear: 1. Pick symbols: write each quantity as a variable (r, h, V, x, y, s, L, etc.) and label the one you’re asked for with d(·)/dt (dx/dt, dV/dt). 2. Find the relation: write an equation linking the variables (Pythagorean, V = πr^2h, similar triangles, area formula, etc.). This uses CHA-3.D knowledge (chain rule + implicit differentiation). 3. Differentiate both sides with respect to time t (use chain rule, product/quotient as needed) so every derivative is a rate like dr/dt, dh/dt. 4. Solve algebraically for the requested rate and plug in the given numbers (including units!). Check units and sign (positive vs negative). 5. State the answer with units and a short sentence interpreting it. Common traps: don’t confuse “how fast x is changing” with x itself; if they give dx/dt but ask for dy/dt you must use the relation. For extra practice and worked examples (ladder, cone, balloon), see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2) and try problems from Fiveable’s practice set (https://library.fiveable.me/practice/ap-calculus).

Use the quotient rule whenever the equation that relates the quantities is a quotient of two functions that both depend on time and you can’t (or don’t want to) simplify it into a product first. In related rates you always differentiate implicitly with respect to t (chain rule/CED CHA-3.D). If your relation is F(t) = u(t)/v(t), apply d/dt[u/v] = (v du/dt − u dv/dt)/v^2 so every occurrence of u and v gets a corresponding du/dt and dv/dt. Examples where this shows up: a density = mass/area where both mass and area change, or a rate expressed as a ratio of two changing lengths. Quick tips: - Try algebraic simplification first (rewrite u/v as u·v⁻¹ and use product + chain rule)—same result but sometimes cleaner. - Always label which variable depends on t and include units (AP free-response expects clear notation like dx/dt). For a quick refresher on solving related-rates problems and examples, see the Topic 4.4 study guide (https://library.fiveable.me/ap-calculus/unit-4/intro-related-rates/study-guide/WxyKc3lpYx3sCEzkH9y2). For lots of practice problems, check Fiveable’s practice page (https://library.fiveable.me/practice/ap-calculus).