9.4 Defining and Differentiating Vector-Valued Functions

Take the derivative component-wise. If r(t) = , then r′(t) = , and r″(t) = . r′(t) is the velocity vector; |r′(t)| is the speed. The unit tangent T(t) = r′(t)/|r′(t)| when |r′(t)| ≠ 0. You can use the same rules as for scalar functions: sum/difference, constant multiple, product and chain rules apply component-wise (and via vector product/inner-product rules when those appear). Differentiability of r means each component is differentiable (limit definition extends to vectors). For motion problems, velocity = r′, acceleration = r″, and total distance = ∫ |r′(t)| dt. This matches the AP CED objective CHA-3.H (calculate derivatives of vector-valued functions). For examples and extra practice, see the topic study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and Tons of practice questions at (https://library.fiveable.me/practice/ap-calculus).

If r(t) = (or in 2D), differentiate component-wise: - Velocity (first derivative): v(t) = r′(t) = . - Acceleration (second derivative): a(t) = r″(t) = . - Speed: |v(t)| = sqrt[(x′(t))^2 + (y′(t))^2 + (z′(t))^2]. - Unit tangent: T(t) = v(t) / |v(t)| (when |v(t)| ≠ 0). You can justify these with the vector limit definition r′(t) = lim_{h→0} (r(t+h)-r(t))/h—the limit splits into limits of components. Product and chain rules work the same way (apply component-wise and use vector-scalar rules). For AP BC, you’ll be expected to compute v, a, speed, and T and use them in motion/context problems (CED CHA-3.H.1). For review see the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Use vector derivatives whenever your function gives a position or quantity as a vector of components that depend on a parameter (usually t). If r(t) = then r′(t) = —you differentiate component-wise. That vector derivative is the velocity; r″(t) is acceleration. Use regular (single-variable) derivatives when you have a scalar function y = f(x) or when you need dy/dx from parametric pieces: dy/dx = (dy/dt)/(dx/dt). Quick rules to pick: - Position given as r(t) → take vector derivatives (velocity/acceleration, tangent vector, speed = |r′(t)|). - Scalar quantity (area, y(x), one variable) → use regular derivatives. - For mixed expressions use vector product/chain rules analogs (they extend from single-variable rules; see CHA-3.H in the CED). If you want practice, review Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and try problems on the AP practice page (https://library.fiveable.me/practice/ap-calculus).

Think of r(t) = <3t^2, sin t, e^t> as a position vector with three component functions. Differentiate component-wise (AP CED CHA-3.H: extend single-variable rules to vectors). Step-by-step: - x(t) = 3t^2 → x'(t) = 6t - y(t) = sin t → y'(t) = cos t - z(t) = e^t → z'(t) = e^t So the velocity vector v(t) = r'(t) = <6t, cos t, e^t>. If you need acceleration, differentiate again component-wise: - x''(t) = 6 - y''(t) = -sin t - z''(t) = e^t So a(t) = r''(t) = <6, -sin t, e^t>. If asked for speed, compute |v(t)| = sqrt((6t)^2 + (cos t)^2 + (e^t)^2). No product/chain tricks were needed here—just apply basic derivative rules to each component. For quick review on AP vector derivatives check the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and hit up practice problems (https://library.fiveable.me/practice/ap-calculus).

Yes—you differentiate a vector-valued function component-wise. If r(t) = , then r′(t) = (this follows from the limit definition applied to each component). That derivative is the velocity vector; r″(t) = is acceleration. Speed = ||r′(t)||, and the unit tangent T(t) = r′(t)/||r′(t)|| when the curve is smooth (||r′(t)|| ≠ 0). All the single-variable rules extend component-wise: product and chain rules apply (e.g., d/dt[f(t)u(t)] = f′(t)u(t)+f(t)u′(t) for scalar f and vector u), and you can compute second derivatives by differentiating components again. For AP BC, be ready to use these ideas for velocity/acceleration, tangent vectors, and smoothness (Topic 9.4). For a quick refresher, see the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM). For more practice, check Fiveable’s unit and practice question pages (https://library.fiveable.me/ap-calculus/unit-9, https://library.fiveable.me/practice/ap-calculus).

Short answer: the processes are the same idea but you work component-wise and keep vector rules in mind. Details: a vector-valued function r(t) = ⟨x(t), y(t), z(t)⟩ is differentiable exactly when each component is differentiable. You differentiate each component: r′(t)=⟨x′(t), y′(t), z′(t)⟩ (limit definition and basic rules extend to vectors). Interpretations: r(t) is position, r′(t) is velocity (vector), |r′(t)| is speed (scalar), and r″(t) is acceleration. For curvature/tangent work use the velocity to get the tangent vector and normalize for the unit tangent T(t)=r′(t)/|r′(t)| when |r′(t)|≠0. Product and chain rules hold but apply to components (and for dot/cross products use their product rules). A “smooth” curve means r′(t) is continuous and not zero. For AP BC review, see Topic 9.4 in the CED (CHA-3.H) and the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM). For more practice, check the AP Calc practice bank (https://library.fiveable.me/practice/ap-calculus).

On the AP, solving vector-derivative problems is mostly mechanical—do everything component-wise and keep track of units. Steps you’ll use every time: - Write r(t) = and differentiate each component: r′(t)= (velocity). CHA-3.H in the CED. - Speed = |r′(t)| = sqrt(x′(t)^2+...). If they ask for unit tangent, T(t)=r′(t)/|r′(t)|. - Acceleration = r″(t) = derivative of r′(t) component-wise. Use product and chain rules as with scalar functions. - For second derivative dy/dx or d^2y/dx^2 in parametric form use dy/dx = (y′/x′) and d^2y/dx^2 = (x′y″ − y′x″)/(x′)^3. - Check smoothness: x′ and y′ not both zero when dividing by x′ or forming unit tangent. On the exam, always show the component work and include units/context when motion is involved. For review see the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Yes—the same derivative ideas carry over, but applied component-wise and with a few vector-specific forms. - Component-wise: if r(t) = , then r′(t) = (this is how you get velocity/acceleration). (CED: CHA-3.H) - Product rule: for a scalar p(t) and vector r(t), (p r)′ = p′ r + p r′. For dot or cross products of two vector functions u(t), v(t): (u·v)′ = u′·v + u·v′ and (u×v)′ = u′×v + u×v′. - Chain rule: if r is a vector function of a scalar s and s = s(t), then d/dt[r(s(t))] = r′(s(t))·s′(t) (differentiate the vector components and multiply by s′(t)). - Limits/derivative definition and smoothness carry over; always check components for differentiability. For more AP-aligned review, see the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and Unit 9 overview (https://library.fiveable.me/ap-calculus/unit-9). Practice problems are at (https://library.fiveable.me/practice/ap-calculus).

Usually your calculator is “weird” because it’s treating the whole vector like one expression instead of differentiating component-wise. For a vector r(t) = you must compute r′(t) = (velocity) and r″(t) = (acceleration). Common calculator mistakes: - Entering angle brackets or commas incorrectly so the device thinks it’s one symbolic object. - Using a numeric derivative routine (d/dx) that approximates slopes at a point instead of giving symbolic component derivatives. - Wrong variable or mode (degree vs. radian for trig in components). - Forgetting chain/product rules when components have nested functions. On the AP exam you’re expected to do component-wise differentiation and use product/chain rules as needed (CED CHA-3.H.1). If you want step practice, check the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and hundreds of practice problems (https://library.fiveable.me/practice/ap-calculus).

Vector derivatives work just like single-variable derivatives but done component-wise. If r(t) = ⟨x(t), y(t), z(t)⟩, then r′(t) = ⟨x′(t), y′(t), z′(t)⟩ (limit definition applies component-wise). r′ is the velocity vector; |r′(t)| is speed. r″(t) = ⟨x″, y″, z″⟩ is acceleration. The tangent direction is r′(t); the unit tangent T(t) = r′(t)/|r′(t)| when |r′(t)| ≠ 0 (smooth curve). Standard rules extend: product rule and chain rule hold for vector-valued functions (apply components or use vector rules). For second derivative d²y/dx² of a parametric curve use (d/dt (dy/dx))/(dx/dt) = (y″ x′ − y′ x″)/(x′)³. AP focus: be able to compute these, interpret velocity/acceleration, and use limit definition when needed (CED CHA-3.H). For step-by-step examples and AP-style practice, see the topic study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and practice problems (https://library.fiveable.me/practice/ap-calculus).

You need the derivative of a parametric curve written as a vector whenever you want the curve’s instantaneous motion or direction—basically anytime you care about velocity, tangent lines, speed, or acceleration. If r(t) = ⟨x(t), y(t), z(t)⟩ then r′(t) = ⟨x′(t), y′(t), z′(t)⟩ (component-wise differentiation). Use r′(t) to: - find the tangent line: point r(t0) and direction r′(t0); - get speed: |r′(t)| (for arc length or how fast the particle moves); - get acceleration: r″(t) for concavity/force problems; - compute dy/dx for a parametric plane curve: (dy/dt)/(dx/dt); - check smoothness: r′(t) ≠ 0 on an interval. These are exactly the CHA-3.H skills on the AP BC CED. For a short study guide and examples on computing these derivatives and using them in motion problems, see the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM). For extra practice, Fiveable has lots of AP Calc problems (https://library.fiveable.me/practice/ap-calculus).

r′(t) means the derivative of a vector-valued position function r(t) = ⟨x(t), y(t), z(t)⟩ (or ⟨x(t), y(t)⟩) taken component-wise. In vector notation r′(t) = ⟨x′(t), y′(t), z′(t)⟩. Interpretation: r′(t) is the velocity vector (tangent vector) to the parametric curve at time t. Its magnitude |r′(t)| is the speed. You find r′(t) using the same derivative rules you know (sum, product, chain) applied to each component—e.g., differentiate x(t), y(t) separately. If asked for a unit tangent, T(t) = r′(t)/|r′(t)| once r′(t) ≠ 0. This is exactly what the AP CED expects for CHA-3.H: extend single-variable derivative rules to vector functions and identify velocity/acceleration (r″(t)). For a quick review see the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Yes—when you differentiate a vector-valued function r(t) = you differentiate each component separately: r′(t) = . The limit/derivative rules from single-variable calculus extend component-wise (CHA-3.H.1). So velocity = r′(t), acceleration = r″(t) found by differentiating components again, and speed = |r′(t)|. Two quick reminders: - If you want dy/dx for a parametric curve, use (dy/dt)/(dx/dt)—not the vector derivative. - If dx/dt and dy/dt are both 0 at a point, the curve may not be smooth there (tangent direction might be undefined). This is exactly what the CED means by “component-wise differentiation.” For a short study refresher see the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM). For extra practice, try problems on the Unit 9 page (https://library.fiveable.me/ap-calculus/unit-9) or the AP practice bank (https://library.fiveable.me/practice/ap-calculus).

Short answer: yes—differentiate vector-valued functions component-wise. If r(t) = ⟨x(t), y(t), z(t)⟩ then r′(t) = ⟨x′(t), y′(t), z′(t)⟩. That’s the main “shortcut” the AP expects (CED CHA-3.H: extend single-variable rules to vectors). What else to know (quick list): - Velocity = r′(t); acceleration = r″(t); speed = |r′(t)|; tangent vector = r′(t); unit tangent = r′(t)/|r′(t)| when |r′| ≠ 0. - Product rule: d/dt [f(t)·r(t)] = f′(t)r(t) + f(t)r′(t) for scalar f. - Chain rule: for r(u(t)), d/dt r = r′(u(t))·u′(t) (apply to each component). - Use usual single-variable rules on each component (power, trig, exp, etc.). If you want practice, the topic study guide walks through examples (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and there are 1000+ practice problems at (https://library.fiveable.me/practice/ap-calculus).

They're the same thing: r'(t) and dr/dt both mean "the derivative of the vector-valued position function r(t)." For a position vector r(t) = , you can read either notation as the componentwise derivative r'(t) = dr/dt = . On the AP CED this is exactly extending single-variable derivative rules to vectors (CHA-3.H.1): interpret r'(t) as the velocity vector, r''(t) as acceleration, and |r'(t)| as speed. You can define r'(t) with the same limit definition used for real functions, and use the same product and chain rules componentwise. For quick practice and AP-style problems on differentiating vector functions, see the Topic 9.4 study guide (https://library.fiveable.me/ap-calculus/unit-9/defining-differentiating-vector-valued-functions/study-guide/z7ZmELI67oVaf9UpfEEM) and more practice questions at (https://library.fiveable.me/practice/ap-calculus).