The Product Rule - AP Calculus AB

1. A function is $F(t)=(t^2-4)\,\sin(t)$. What is $F'(t)$?
   Answer: $2t\sin(t)+(t^2-4)\cos(t)$ F(t) = (t² - 4) sin(t) requires product: 2t sin(t) + (t² - 4) cos(t). Basic trig product. Error: Omitting cos term, like B. Strategy: Simple sin products always include cos in second term.
2. A model is $H(x)=(x^3)\,\cos(x^2)$. What is $H'(x)$?
   Answer: $3x^2\cos(x^2)-2x^4\sin(x^2)$ The product rule states that the derivative of a product of two functions, f(x) and g(x), is f'(x)g(x) + f(x)g'(x). Here, H(x) is the product of x³ and cos(x²), so the product rule is required to differentiate it correctly. First, differentiate x³ to get 3x², multiply by cos(x²), then add x³ times the derivative of cos(x²), which involves the chain rule to yield -sin(x²) · 2x. A tempting incorrect shortcut might be to treat it as a simple power or forget the chain rule in the second term, leading to errors like choice B. Instead, always apply the product rule fully and compute each derivative accurately. For any product of functions, identify the two parts, compute their derivatives separately, and combine using the product rule formula to ensure correctness.
3. A rate is $q(t)=(t^2+2t+1)\,\sin(t^2)$. What is $q'(t)$?
   Answer: $(2t+2)\sin(t^2)+(t^2+2t+1)\cdot 2t\cos(t^2)$ q(t) = (t² + 2t + 1) sin(t²) requires product: (2t + 2) sin(t²) + (t² + 2t + 1) (2t cos(t²)). Chain for sin(t²). Error: Missing chain, like A. Strategy: Composite sin needs chain in product.
4. A function is $U(t)=(t^2+2t)\,\cos(5t)$. What is $U'(t)$?
   Answer: $(2t+2)\cos(5t)-5(t^2+2t)\sin(5t)$ U(t) = (t² + 2t) cos(5t) needs product: (2t + 2) cos(5t) - 5 (t² + 2t) sin(5t). Quadratic-composed cos. Composed: -5 sin(5t). Error: omit chain, like choice B. Include. For composed trig with polys.
5. A rate model uses $w(x)=(2x+1)\,3^{x}$. What is $w'(x)$?
   Answer: $2\cdot 3^{x}+(2x+1)\,3^{x}\ln 3$ The function $w(x) = (2x+1) \cdot 3^x$ requires the product rule for a product involving an exponential function. Let $u = 2x+1$ with $u' = 2$, and $v = 3^x$ with $v' = 3^x \ln 3$. Applying the product rule: $w'(x) = 2 \cdot 3^x + (2x+1) \cdot 3^x \ln 3$. This gives us $w'(x) = 2 \cdot 3^x + (2x+1)3^x \ln 3$. A common mistake with exponential functions like $3^x$ is forgetting the chain rule factor of $\ln 3$. Remember that $(a^x)' = a^x \ln a$ for any positive constant $a \neq 1$. When products involve exponential functions, use the product rule just as with any other functions, but be careful with the exponential derivatives.
6. A pressure function is $P(x)=(x^2+1)\arctan(x)$. What is $P'(x)$?
   Answer: $2x\arctan(x)+\dfrac{x^2+1}{1+x^2}$ P(x) = (x² + 1) arctan(x) uses product: 2x arctan(x) + (x² + 1) (1/(1 + x²)). Arctan derivative. Shortcut: Forgetting second term, like C. Strategy: Inverse trig products include their derivatives.
7. A profit function is $p(x)=(2x-1)\sin(3x)$. What is $p'(x)$?
   Answer: $2\sin(3x)+(2x-1)\cdot 3\cos(3x)$ p(x) = (2x - 1) sin(3x) needs the product rule: 2 sin(3x) + (2x - 1) (3 cos(3x)), incorporating the chain rule for sin(3x). Without chain rule, one might get choice A. A shortcut is omitting the chain, leading to errors. Proper method includes it. Strategy: For products with composite trig, combine product and chain rules.
8. A cost function is $C(x)=(\sqrt{x})(x^2+4)$. What is $C'(x)$ for $x>0$?
   Answer: $\dfrac{1}{2\sqrt{x}}(x^2+4)+\sqrt{x}(2x)$ C(x) is a product of √x and (x² + 4), so use the product rule: (1/(2√x))(x² + 4) + √x (2x). This accounts for both parts changing with x. A tempting shortcut is to treat √x as constant or miscompute its derivative, leading to just √x (2x) as in D. Others might forget the chain rule in the square root derivative. The correct method combines them properly. Always verify if the function is a product and differentiate each factor accurately for the strategy.
9. A function is $Q(x)=(x^2-4x+4)\,e^{-x}$. What is $Q'(x)$?
   Answer: $(2x-4)e^{-x}-(x^2-4x+4)e^{-x}$ Q(x) = (x² - 4x + 4) $e^{-x}$ needs product: (2x - 4)$e^{-x}$ - (x² - 4x + 4)$e^{-x}$. For quadratic-exponential. Exponential: -$e^{-x}$. Tempting: only first, choice B. Full rule. Apply to polys with exponentials.
10. A function is $d(x)=(x^2-1)\,\arcsin(x)$. What is $d'(x)$?
    Answer: $2x\arcsin(x)+\dfrac{x^2-1}{\sqrt{1-x^2}}$ Differentiate d(x) = (x² - 1) arcsin(x) with product rule: 2x arcsin(x) + (x² - 1)(1/√(1 - x²)). Required for product of polynomial and inverse trig. Arcsin derivative is 1/√(1 - x²). Incorrect shortcut: omitting product, like choice D. Include all terms. Identify products with inverse functions and apply rule consistently.
11. A velocity is $v(t)=(t^3+5)\,(t^{-1})$. What is $v'(t)$ for $t\ne0$?
    Answer: $3t^2\cdot t^{-1}+(t^3+5)(-t^{-2})$ v(t) = (t³ + 5) $t^{-1}$ is product: 3t² $t^{-1}$ + (t³ + 5) ($-t^{-2}$). Handles negative exponent. Error: Treating as quotient incorrectly. Strategy: Rewrite powers and apply product if multiplied.
12. A volume rate is $V(x)=(x^2-3x)\,\ln(x^2+1)$. What is $V'(x)$?
    Answer: $(2x-3)\ln(x^2+1)+\dfrac{2x(x^2-3x)}{x^2+1}$ V(x) = (x² - 3x) ln(x² + 1) uses product: (2x - 3) ln(x² + 1) + (x² - 3x) (2x/(x² + 1)). Includes chain for log. Shortcut: Omitting second term, like B. Strategy: Log products with composites require chain in rule.
13. A demand function is $D(p)=(p+1)\,e^{p^2}$. What is $D'(p)$?
    Answer: $e^{p^2}+2p(p+1)e^{p^2}$ D(p) = (p + 1) $e^{p²}$ needs product: 1 · $e^{p²}$ + (p + 1) (2p $e^{p²}$). Chain for exponent. Shortcut: Missing chain, like C. Strategy: Exponential with variable power uses chain in product.
14. A function is $J(x)=(x^2+4x)\,\sec^2(x)$. What is $J'(x)$?
    Answer: $(2x+4)\sec^2(x)+2(x^2+4x)\sec^2(x)\tan(x)$ J(x) is the product of (x² + 4x) and sec²(x), so the product rule gives (2x + 4)sec²(x) + (x² + 4x)(2 sec²(x) tan(x)). This is required due to the multiplication of a polynomial and a trigonometric function with a power. The derivative of sec²(x) uses the chain rule: 2 sec(x) · (sec(x) tan(x)). An incorrect shortcut might be to forget the chain rule or omit one product term, like in choice C. Apply the rule carefully to both parts. When faced with trigonometric powers multiplied by other functions, confirm it's a product and use the rule with any necessary chain applications.
15. A moving object has position $s(t)=(t^2+3)(\sin t)$. What is $s'(t)$?
    Answer: $2t\sin t+(t^2+3)\cos t$ This problem requires the product rule because we have two functions multiplied together: $(t^2+3)$ and $\sin t$. The product rule states that if $h(t) = f(t) \cdot g(t)$, then $h'(t) = f'(t) \cdot g(t) + f(t) \cdot g'(t)$. Here, $f(t) = t^2+3$ with $f'(t) = 2t$, and $g(t) = \sin t$ with $g'(t) = \cos t$. Applying the product rule: $s'(t) = 2t \cdot \sin t + (t^2+3) \cdot \cos t$. A common mistake is to differentiate each factor separately without adding them, which would give just $(t^2+3)\cos t$. Remember: when you see multiplication of two variable expressions, use the product rule—differentiate the first times the second, plus the first times the derivative of the second.
16. A force is modeled by $F(x)=\tan(x)\,(x^2)$. What is $F'(x)$?
    Answer: $x^2\sec^2(x)+2x\tan(x)$ F(x) = tan(x) x² is a product, requiring the rule: sec²(x) x² + tan(x) (2x). This captures trig and polynomial changes. A common shortcut is differentiating only the tan, giving x² sec²(x) as in D. Others might square the derivative incorrectly. Full application is key. Strategy: Spot trig-polynomial multiples and apply product rule to both terms.
17. A signal is $S(t)=(e^{-t})(\sin t)$. What is $S'(t)$?
    Answer: $-e^{-t}\sin t+e^{-t}\cos t$ S(t) = $e^{-t} \sin t$ is a product: $(-e^{-t} \sin t) + e^{-t} \cos t$, or $-e^{-t} \sin t + e^{-t} \cos t$. Note chain for exponential. Error: Forgetting chain, like B. Rule with chain is correct. Strategy: Exponential-trig products need both rules.
18. A function is $b(x)=(x^2+1)\,\sin^2(x)$. What is $b'(x)$?
    Answer: $2x\sin^2(x)+(x^2+1)\cdot 2\sin(x)\cos(x)$ For b(x) = (x² + 1) sin²(x), the product rule yields 2x sin²(x) + (x² + 1)(2 sin(x) cos(x)). It's required as a product of a polynomial and a squared sine. The derivative of sin²(x) is 2 sin(x) cos(x) via chain rule. Tempting shortcuts include forgetting the chain or one term, as in choice D. Always include both. Check for product form; if present, use the rule and chain where needed for transferable accuracy.
19. A motion model is $m(x)=(\cos x)(\ln x)$. What is $m'(x)$ for $x>0$?
    Answer: $-\sin x\ln x+\dfrac{\cos x}{x}$ m(x) = cos(x) ln(x) uses product: -sin(x) ln(x) + cos(x) (1/x). Combines trig and log. Shortcut: Omitting log derivative, like C. Strategy: Trig-log products differentiate each.
20. A model is $k(x)=(x^2+1)\,\ln(2x+1)$. What is $k'(x)$?
    Answer: $2x\ln(2x+1)+\dfrac{2(x^2+1)}{2x+1}$ k(x) = (x² + 1) ln(2x + 1) uses product: 2x ln(2x + 1) + (x² + 1) (2/(2x + 1)). Chain for log. Shortcut: Missing chain, like E. Strategy: Log with linear argument needs chain.

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