For f(x) = (x³ - 1)/ln x, we apply the quotient rule with numerator x³ - 1 (derivative: 3x²) and denominator ln x (derivative: 1/x). The quotient rule yields f'(x) = [3x²·ln x - (x³ - 1)·(1/x)]/(ln x)². The trickiest part is handling the derivative of ln x correctly—it's 1/x, not just 1. A common error is reversing the subtraction order in the numerator, which would change the sign of the entire derivative. Remember the mnemonic "lo d(hi) - hi d(lo), over lo squared" where lo is the denominator and hi is the numerator.

To differentiate R(t) = eᵗ/(t² - 4), we identify the numerator as eᵗ with derivative eᵗ, and the denominator as t² - 4 with derivative 2t. Applying the quotient rule: R'(t) = [eᵗ(t² - 4) - eᵗ(2t)]/(t² - 4)². Notice that eᵗ appears in both terms of the numerator, which we could factor out as eᵗ[(t² - 4) - 2t]/(t² - 4)², simplifying to eᵗ(t² - 2t - 4)/(t² - 4)². The key insight is maintaining the correct order: "bottom·(top)' - top·(bottom)'" to avoid sign errors. Always square the denominator in your final answer.

For s(t) = (t cos t)/(t² + 1), we first find the derivative of the numerator using the product rule: d/dt(t cos t) = cos t + t(-sin t) = cos t - t sin t. The denominator's derivative is 2t. Applying the quotient rule: s'(t) = [(cos t - t sin t)(t² + 1) - (t cos t)(2t)]/(t² + 1)². The challenge here is correctly applying the product rule within the quotient rule—many students forget to differentiate t cos t as a product. Always identify composite functions and apply the appropriate rules in sequence, working from the inside out.

To find h'(x) for h(x) = (2x + 5)/x², we have numerator 2x + 5 with derivative 2, and denominator x² with derivative 2x. The quotient rule gives h'(x) = [2·x² - (2x + 5)·2x]/(x²)² = [2x² - 2x(2x + 5)]/x⁴. Expanding the second term: 2x² - 4x² - 10x = -2x² - 10x = -2x(x + 5). A common mistake is writing the denominator as x² instead of x⁴, forgetting to square it. Remember that (x²)² = x⁴, and always verify your denominator's exponent after applying the quotient rule.

For w(x) = $\sqrt{x^2 + 1}/x$, the quotient rule gives $[g'(x) h(x) - g(x) h'(x)] / [h(x)]^2$. g(x) = $\sqrt{x^2 + 1}$, g'(x) = $x / \sqrt{x^2 + 1}$, h(x) = $x$, h'(x) = $1$, resulting in $[ (x / \sqrt{x^2 + 1}) x - \sqrt{x^2 + 1} \cdot 1 ] / x^2$, choice A. Errors include wrong subtraction order. Omitting square leads to choice C. Root derivative is tricky. Practice rewriting functions to simplify before differentiating.

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."

The quotient rule for derivatives applies to functions expressed as g(x)/h(x), where the derivative is [g'(x) h(x) - g(x) h'(x)] / $[h(x)]^2$. For P(x) = sin x / (x² + 1), g(x) = sin x with g'(x) = cos x, and h(x) = x² + 1 with h'(x) = 2x. This yields [cos x (x² + 1) - sin x (2x)] / (x² + 1)², corresponding to choice B. Students often err by adding instead of subtracting in the numerator, as in choice A, or forgetting to square the denominator, like in choice C. Misapplying the rule by reversing the subtraction would also lead to incorrect signs. A useful strategy is to label the components g, h, g', h' on paper and double-check the formula's structure before computation.

To differentiate f(x) = √x/sin(x), recognize that g(x) = √x = x^(1/2) with g'(x) = 1/(2√x), and h(x) = sin(x) with h'(x) = cos(x). The quotient rule gives f'(x) = [(1/(2√x))·sin(x) - √x·cos(x)]/[sin(x)]². The key challenge is correctly differentiating √x to get 1/(2√x), not 1/√x or √x/2. The denominator must be [sin(x)]², which equals sin²(x), not just sin(x). A helpful strategy is to rewrite radicals as fractional exponents before differentiating: √x = x^(1/2), so d/dx[x^(1/2)] = (1/2)x^(-1/2) = 1/(2√x).

The quotient rule for S(x) = g(x)/h(x) is [g'(x) h(x) - g(x) h'(x)] / $[h(x)]^2$. With g(x) = x cos x, g'(x) = cos x - x sin x (by product rule), h(x) = x + 1, h'(x) = 1, this gives [(cos x - x sin x)(x + 1) - (x cos x)(1)] / (x + 1)², matching choice A. Errors often arise from reversing the numerator subtraction, changing the sign. Forgetting to square the denominator results in choice C. Mishandling the product rule for g'(x) is another common issue. To build proficiency, break down composite derivatives first and then apply the quotient rule carefully.

The quotient rule is used to find the derivative of a function that is the ratio of two differentiable functions, stating that if R(t) = g(t)/h(t), then R'(t) = [g'(t) h(t) - g(t) h'(t)] / $[h(t)]^2$. Here, g(t) = t² + 3t with g'(t) = 2t + 3, and h(t) = t - 4 with h'(t) = 1. Applying the rule gives [(2t + 3)(t - 4) - (t² + 3t)(1)] / (t - 4)², which matches choice A. A common error is reversing the numerator to g h' - g' h, resulting in the negative of the correct expression. Another frequent mistake is not squaring the denominator, leading to an incorrect form like choice C. To master the quotient rule, always identify and compute g, h, g', and h' separately before substituting into the formula, ensuring the subtraction order and denominator are correct.