For q(x) = ln(x+1)/x, the quotient rule is [g'(x) h(x) - g(x) h'(x)] / $[h(x)]^2$, with g(x) = ln(x+1), g'(x) = 1/(x+1), h(x) = x, h'(x) = 1. This yields [ (1/(x+1)) x - ln(x+1) (1) ] / x², choice A. A typical error is adding terms in the numerator instead. Not squaring the denominator leads to choice C. Logarithmic derivative mishaps occur often. To enhance accuracy, verbalize the rule as you apply it to each problem.

To find $S'(x)$ for $S(x) = \dfrac{x \cos x}{x + 1}$, first find the derivative of the numerator using the product rule: $\dfrac{d}{dx}[x \cos x] = \cos x - x \sin x$. The denominator is $x + 1$ with derivative 1. Applying the quotient rule: $S'(x) = \dfrac{ [ \cos x - x \sin x ] (x + 1) - x \cos x \cdot 1 }{ (x + 1)^2 }$. A critical insight is recognizing that the numerator $x \cos x$ requires the product rule before applying the quotient rule. Students often forget this nested differentiation and incorrectly use just $\cos x$ as the derivative of the numerator. Always identify whether the numerator or denominator requires additional differentiation rules before applying the quotient rule.

To differentiate R(x) = (5x - 1)/(x² + 4), we need the quotient rule with u(x) = 5x - 1 (so u'(x) = 5) and v(x) = x² + 4 (so v'(x) = 2x). Applying the formula gives R'(x) = [5(x² + 4) - (5x - 1)(2x)]/(x² + 4)². A frequent error is to differentiate only the numerator and ignore the denominator, which would incorrectly give just 5/(x² + 4) as in option D. Another mistake is forgetting to square the denominator, resulting in option C. The key insight is that when differentiating a quotient, both parts interact through the quotient rule—you cannot treat them separately. Memorize the pattern: "(derivative of top)(bottom) minus (top)(derivative of bottom), all divided by (bottom)²."

For v(t) = (t³ - 4)/(2t + 1), we have u(t) = t³ - 4 with u'(t) = 3t², and v(t) = 2t + 1 with v'(t) = 2. Applying the quotient rule: v'(t) = [(3t²)(2t + 1) - (t³ - 4)(2)]/(2t + 1)². The coefficient 2 in v'(t) comes from differentiating 2t + 1, which some students forget, thinking the derivative is just 1. Option D shows what happens if you ignore the constant term -4 in the numerator, which is incorrect—every term matters in the quotient rule. The denominator (2t + 1)² confirms we're using the quotient rule correctly, not just differentiating the fraction term by term. Remember: for quotients, use (u'v - uv')/v², where every piece plays a crucial role.

The quotient rule for T(x) = g(x)/h(x) is [g'(x) h(x) - g(x) h'(x)] / $[h(x)]^2$. With g(x) = $x^4$, g'(x) = 4x³, h(x) = x³ + 1, h'(x) = 3x², this results in [4x³ (x³ + 1) - $x^4$ (3x²)] / (x³ + 1)², choice A. A frequent error is using plus instead of minus in the numerator, as in choice B. Not squaring the denominator gives choice C. Power rule mistakes in g' or h' are common. Develop a habit of writing the formula and components side-by-side for error-free application.

The quotient rule for F(t) = g(t)/h(t) gives [g'(t) h(t) - g(t) h'(t)] / $[h(t)]^2$. With g(t) = t² sin t, g'(t) = 2t sin t + t² cos t (product rule), h(t) = t² + 1, h'(t) = 2t, resulting in [(2t sin t + t² cos t)(t² + 1) - (t² sin t)(2t)] / (t² + 1)², choice A. Reversing subtraction is a common numerator error. Forgetting to square denominator gives choice C. Complex g' requires careful product rule. Practice by differentiating parts separately then combining for reliability.

For a(t) = sin(2t)/(t² - 4), the quotient rule is [g'(t) h(t) - g(t) h'(t)] / $[h(t)]^2$. g(t) = sin(2t), g'(t) = 2 cos(2t), h(t) = t² - 4, h'(t) = 2t, giving [2 cos(2t) (t² - 4) - sin(2t) (2t)] / (t² - 4)², choice A. Reversing subtraction is frequent. Not squaring denominator leads to choice C. Chain rule for sin(2t) is essential. Apply the rule consistently by memorizing its verbal form.

The quotient rule for I(x) = (3x - 1)/√x is [g'(x) h(x) - g(x) h'(x)] / $[h(x)]^2$. g(x) = 3x - 1, g'(x) = 3, h(x) = √x, h'(x) = 1/(2√x), yielding [3 √x - (3x - 1) (1/(2√x))] / x, since [√x]² = x, choice A but denominator is $(√x)^2$ = x, yes. Common errors: wrong numerator order. Omitting square gives choice C. Root in denominator needs care. To master, rewrite quotients as products and use product rule for cross-verification when possible.

The quotient rule applies to B(x) = cos x / (2x - 1) as [g'(x) h(x) - g(x) h'(x)] / $[h(x)]^2$. g(x) = cos x, g'(x) = -sin x, h(x) = 2x - 1, h'(x) = 2, yielding [-sin x (2x - 1) - cos x (2)] / (2x - 1)², matching choice A. Errors often stem from wrong numerator signs or order. Forgetting to square denominator results in choice C. Trigonometric derivatives need precision. To transfer skills, compare your result with software or graphs for verification.

The quotient rule for D(p) = (p + 4)/(p² - 1) gives [g'(p) h(p) - g(p) h'(p)] / $[h(p)]^2$. g(p) = p + 4, g'(p) = 1, h(p) = p² - 1, h'(p) = 2p, resulting in [1 (p² - 1) - (p + 4)(2p)] / (p² - 1)², choice A. Typical errors include adding in numerator. Omitting square on denominator gives choice C. Factorable denominators might tempt simplification too soon. Use the rule's structure to guide computation in similar rational functions.