This problem requires the quotient rule to find the derivative of R(t) = (t² + 3t)/(t - 1). The quotient rule states that if f(x) = g(x)/h(x), then f'(x) = [g'(x)·h(x) - g(x)·h'(x)]/[h(x)]². Here, g(t) = t² + 3t with g'(t) = 2t + 3, and h(t) = t - 1 with h'(t) = 1. Applying the formula: R'(t) = [(2t + 3)(t - 1) - (t² + 3t)(1)]/(t - 1)². Choice B incorrectly reverses the order in the numerator, which would give a different result. Remember: in the quotient rule, it's always "derivative of top times bottom minus top times derivative of bottom, all over bottom squared."

To find g'(x) for g(x) = $\dfrac{\ln x}{x + 2}$, we use the quotient rule with numerator $\ln x$ (derivative: $\dfrac{1}{x}$) and denominator $x + 2$ (derivative: 1). Applying the quotient rule formula: g'(x) = $\dfrac{ \left( \dfrac{1}{x} \right) (x + 2) - (\ln x)(1) }{ (x + 2)^2 }$. This simplifies to $\dfrac{ \left( \dfrac{1}{x} \right) (x + 2) - \ln x }{ (x + 2)^2 }$. Choice B incorrectly adds instead of subtracts $\ln x$ in the numerator, violating the quotient rule's subtraction requirement. When using the quotient rule, always maintain the correct order and operation: it's subtraction, not addition, between the two products in the numerator.

Finding w'(x) for w(x) = tan x/x requires the quotient rule with numerator tan x (derivative: sec² x) and denominator x (derivative: 1). Applying the quotient rule formula: w'(x) = [(sec² x)·x - (tan x)·1]/x² = (x sec² x - tan x)/x². Choice C incorrectly reverses the order of subtraction in the numerator, which would change the sign and give an incorrect result. The quotient rule's mnemonic "low d-high minus high d-low, all over low squared" helps remember the correct order, where maintaining the subtraction order is essential for the correct derivative.

This problem requires the quotient rule to find the derivative of the rate function F(x) = (√x + 1)/(x - 3). The quotient rule states that if f(x) = num(x)/den(x), then f'(x) = [num'(x) den(x) - num(x) den'(x)] / [den(x)]². Here, num(x) = √x + 1 with num'(x) = 1/(2√x), and den(x) = x - 3 with den'(x) = 1. Applying the rule gives [(1/(2√x))(x - 3) - (√x + 1)·1] / (x - 3)², matching the correct choice. A tempting distractor like choice D fails by omitting the square on the denominator, which is a key part of the quotient rule. Remember, a transferable strategy for the quotient rule is to always write 'low d-high minus high d-low over low squared' to keep the order and signs correct.

This problem requires the quotient rule to find the derivative of the average cost function A(x) = (5x - 2)/(x² + 1). The quotient rule states that if f(x) = num(x)/den(x), then f'(x) = [num'(x) den(x) - num(x) den'(x)] / [den(x)]². Here, num(x) = 5x - 2 with num'(x) = 5, and den(x) = x² + 1 with den'(x) = 2x. Applying the rule gives [5(x² + 1) - (5x - 2)(2x)] / (x² + 1)², matching the correct choice. A tempting distractor like choice A fails because it uses a plus sign instead of a minus in the numerator, reversing the subtraction required by the rule. Remember, a transferable strategy for the quotient rule is to always write 'low d-high minus high d-low over low squared' to keep the order and signs correct.

The simplified score s(x) = (x - 2) / (x² - 9) is differentiated using the quotient rule. Let u = x - 2 with u' = 1 and v = x² - 9 with v' = 2x. Applying the rule results in [1 (x² - 9) - (x - 2) 2x] / (x² - 9)², matching choice A. The numerator simplifies to x² - 9 - 2x(x - 2) for further analysis if needed. Choice C incorrectly subtracts in reverse, yielding the opposite sign. Always compute the denominator squared accurately to avoid power errors in quotient rule problems.

To differentiate g(x) = (x³ - 1)/(x² + 4), we use the quotient rule. The quotient rule formula is [f'(x)·g(x) - f(x)·g'(x)]/[g(x)]² where f is the numerator and g is the denominator. The numerator (x³ - 1) has derivative 3x², and the denominator (x² + 4) has derivative 2x. Substituting: g'(x) = [3x²(x² + 4) - (x³ - 1)(2x)]/(x² + 4)². Choice E reverses the order of subtraction in the numerator, which is incorrect—the quotient rule specifically requires numerator' × denominator minus numerator × denominator'. Always maintain the correct order: (top)' × (bottom) - (top) × (bottom)'.

Finding C'(t) for C(t) = eᵗ/(t + 2) requires the quotient rule. The quotient rule states that for f/g, the derivative is (f'g - fg')/g². Here, the numerator eᵗ has derivative eᵗ, and the denominator (t + 2) has derivative 1. Applying the formula: C'(t) = [eᵗ(t + 2) - eᵗ(1)]/(t + 2)² = [eᵗ(t + 2) - eᵗ]/(t + 2)². Choice C shows the common mistake of forgetting to square the denominator in the quotient rule. To avoid errors, always write the squared denominator first, then work on the numerator calculation.

Finding m'(x) for m(x) = (x² + 1)/x³ requires the quotient rule. The quotient rule states (u/v)' = (u'v - uv')/v². The numerator (x² + 1) has derivative 2x, and the denominator x³ has derivative 3x². Applying: m'(x) = [2x·x³ - (x² + 1)·3x²]/(x³)² = [2x⁴ - 3x⁴ - 3x²]/x⁶ = [-x⁴ - 3x²]/x⁶. Choice C forgets to square the denominator, showing x³ instead of (x³)² = x⁶. When using the quotient rule, always remember to square the entire denominator expression in your final answer.

To differentiate $f(x)=\frac{x^2+4x}{\sqrt{x}}$, we use the quotient rule. The quotient rule formula is $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$. With $u(x)=x^2+4x$ giving $u'(x)=2x+4$, and $v(x)=\sqrt{x}$ giving $v'(x)=\frac{1}{2\sqrt{x}}$, we get: $f'(x)=\frac{(2x+4)\sqrt{x}-(x^2+4x)\left(\frac{1}{2\sqrt{x}}\right)}{(\sqrt{x})^2}$. Choice D forgets to square the denominator, writing just $\sqrt{x}$ instead of $(\sqrt{x})^2=x$. The quotient rule requires squaring the entire denominator function—this ensures dimensional consistency and the correct rate of change.