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.

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.

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.

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.

Since h(t) is the product of sin(t) and (t² + 1), the product rule is required: derivative is cos(t)(t² + 1) + sin(t)(2t). This captures the rate of change from both factors. Without it, one might wrongly differentiate only the trigonometric part, yielding just cos(t)(t² + 1) as in C. Another shortcut error is adding derivatives without multiplying, like choice A. The full application avoids these pitfalls. For transferable strategy, scan for multiplication of two non-constant functions and apply the product rule to ensure complete differentiation.

The displacement s(t) = (t - 5) cos(t) is a product, necessitating the product rule: 1 · cos(t) + (t - 5) (-sin(t)), or cos(t) - (t - 5) sin(t). This reflects contributions from both linear and trigonometric terms. A common error is omitting the derivative of the linear part, resulting in just (t - 5)(-sin(t)) as in E. Some might add without the negative sign. Proper application ensures accuracy. To decide, recognize products involving trig functions and apply the rule consistently.

G(x) = (x² + 1) $e^{2x}$ requires the product rule due to the polynomial and exponential product: (2x) $e^{2x}$ + (x² + 1) (2 $e^{2x}$). Note the chain rule for the exponential derivative. A shortcut mistake is forgetting the chain rule, leading to (2x + 1) $e^{2x}$ as in D. Others might omit one term entirely. The full rule combines them. Strategy: Identify exponential products and incorporate chain rule where needed in differentiation.

T(t) = ln(t) (t³ + 2) is a product of logarithmic and polynomial functions, so apply the product rule: (1/t)(t³ + 2) + ln(t)(3t²). This ensures both rates are included. A tempting error is ignoring the log derivative, yielding just ln(t)(3t²) as in E. Some might confuse with quotient rule. Correct use avoids this. For strategy, check for log-polynomial products and systematically differentiate each.

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.

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.