The Quotient Rule - AP Calculus AB

For $q(x)=\dfrac{\ln(x+1)}{x}$ , what is $q'(x)$ ?
Answer: $\dfrac{\frac{1}{x+1}\cdot x-\ln(x+1)\cdot 1}{x^2}$ 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.
In a lab, $C(t)=\dfrac{t^2+3t}{t-2}$ . What is $C'(t)$ ?
Answer: $\dfrac{(2t+3)(t-2)-(t^2+3t)}{(t-2)^2}$ To find C'(t) for C(t) = (t² + 3t)/(t - 2), we apply the quotient rule: 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 - 2 with h'(t) = 1. Substituting into the quotient rule gives C'(t) = [(2t + 3)(t - 2) - (t² + 3t)(1)]/(t - 2)². A common error is forgetting to square the denominator or mixing up the order of subtraction in the numerator. Remember: the quotient rule always has the form "bottom times derivative of top minus top times derivative of bottom, all over bottom squared."
A signal is $s(t)=\dfrac{t\cos t}{t^2+1}$ . What is $s'(t)$ ?
Answer: $\dfrac{(\cos t-t\sin t)(t^2+1)-(t\cos t)(2t)}{(t^2+1)^2}$ 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.
Let $f(x)=\dfrac{x^3-1}{\ln x}$ . What is $f'(x)$ for $x>0$ , $x\ne1$ ?
Answer: $\dfrac{3x^2\ln x-(x^3-1)\cdot\frac{1}{x}}{(\ln x)^2}$ 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.
For a population model, $N(t)=\dfrac{e^t}{t+5}$ . What is $N'(t)$ ?
Answer: $\dfrac{e^t(t+5)-e^t}{(t+5)^2}$ To differentiate a quotient g(t)/h(t), the quotient rule gives [g'(t) h(t) - g(t) h'(t)] / [h(t)]^2. In this case, g(t) = $e^t$ with g'(t) = $e^t$ , and h(t) = t + 5 with h'(t) = 1, so N'(t) = [ $e^t$ (t + 5) - $e^t$ (1)] / (t + 5)², which is choice A. A typical mistake is switching the numerator terms to g h' - g' h, flipping the sign. Forgetting the squared denominator often results in expressions like choice C. Some might simplify prematurely without verifying the rule's application. For consistent success, practice by writing the rule and plugging in values step-by-step, verifying against known derivatives when possible.
For a lens design, $f(x)=\dfrac{x^2+3x}{x-2}$ . What is $f'(x)$ ?
Answer: $\dfrac{(2x+3)(x-2)-(x^2+3x)}{(x-2)^2}$ To find the derivative of f(x) = (x² + 3x)/(x - 2), we apply the quotient rule: if f(x) = u(x)/v(x), then f'(x) = [u'(x)·v(x) - u(x)·v'(x)]/[v(x)]². Here, u(x) = x² + 3x with u'(x) = 2x + 3, and v(x) = x - 2 with v'(x) = 1. Substituting into the quotient rule formula gives f'(x) = [(2x + 3)(x - 2) - (x² + 3x)(1)]/(x - 2)². A common error is forgetting to square the denominator or mixing up the order of subtraction in the numerator. Remember: the quotient rule always has the form "derivative of top times bottom minus top times derivative of bottom, all over bottom squared."
A normalized rate is $N(x)=\dfrac{2x+5}{\sqrt{x^2+1}}$ . What is $N'(x)$ ?
Answer: $\dfrac{2\sqrt{x^2+1}-(2x+5)\cdot\dfrac{x}{\sqrt{x^2+1}}}{(\sqrt{x^2+1})^2}$ For N(x) = (2x + 5)/√(x² + 1), we have g(x) = 2x + 5 with g'(x) = 2, and h(x) = √(x² + 1) with h'(x) = x/√(x² + 1) using the chain rule. The quotient rule gives N'(x) = [2·√(x² + 1) - (2x + 5)·x/√(x² + 1)]/[√(x² + 1)]². Since [√(x² + 1)]² = x² + 1, this matches choice C. A common error is using just (x² + 1) in the denominator without recognizing that we need [h(x)]² = [√(x² + 1)]². Another error might be incorrectly computing h'(x) by forgetting the chain rule for √(x² + 1). When differentiating quotients with composite functions in the denominator, carefully apply the chain rule for h'(x) and remember to square the original denominator function.
For $F(t)=\dfrac{\sqrt{t}}{t+9}$ in a growth model, what is $F'(t)$ ?
Answer: $\dfrac{\frac{1}{2\sqrt{t}}(t+9)-\sqrt{t}}{(t+9)^2}$ To differentiate F(t) = √t/(t + 9), we use g(t) = √t = $t^{1/2}$ with g'(t) = 1/(2√t), and h(t) = t + 9 with h'(t) = 1. The quotient rule gives F'(t) = [(1/(2√t))·(t + 9) - √t·1]/(t + 9)². This matches the form in choice C, where we subtract √t from (1/(2√t))·(t + 9) in the numerator. A common error is using (t + 9) instead of (t + 9)² in the denominator, but the quotient rule requires squaring the entire denominator function. Another error might be adding instead of subtracting in the numerator. When differentiating square roots in quotients, write √t as $t^{1/2}$ first to clearly see that g'(t) = (1/2) $t^{-1/2}$ = 1/(2√t).
For a demand index, $D(x)=\dfrac{e^x}{x^2-3}$ . Find $D'(x)$ .
Answer: $\dfrac{(e^x)(x^2-3)-(e^x)(2x)}{(x^2-3)^2}$ For D(x) = $e^x$ /(x² - 3), we have u(x) = $e^x$ with u'(x) = $e^x$ (since the derivative of $e^x$ is itself), and v(x) = x² - 3 with v'(x) = 2x. The quotient rule yields D'(x) = [( $e^x$ )(x² - 3) - ( $e^x$ )(2x)]/(x² - 3)². Notice that $e^x$ appears in both terms of the numerator, which we could factor out as $e^x$ [(x² - 3) - 2x]/(x² - 3)², though the given answer doesn't show this factored form. Option D incorrectly tries to split the derivative into two separate fractions, which violates the quotient rule. The key insight is that even when the numerator has a simple derivative like $e^x$ , we must still apply the full quotient rule formula. Always use (u'v - uv')/v² for any quotient, regardless of how simple the functions appear.
A device outputs $P(x)=\dfrac{\sin x}{x^2+1}$ . What is $P'(x)$ ?
Answer: $\dfrac{\cos x(x^2+1)-\sin x(2x)}{(x^2+1)^2}$ 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.
A trajectory is $y(t)=\dfrac{t-2}{t^2+5}$ . What is $y'(t)$ ?
Answer: $\dfrac{(t^2+5)-(t-2)(2t)}{(t^2+5)^2}$ To find y'(t) for y(t) = (t - 2)/(t² + 5), we use g(t) = t - 2 with g'(t) = 1, and h(t) = t² + 5 with h'(t) = 2t. The quotient rule gives y'(t) = [1·(t² + 5) - (t - 2)·2t]/(t² + 5)². Expanding the numerator: (t² + 5) - 2t(t - 2) = t² + 5 - 2t² + 4t = -t² + 4t + 5 = (t² + 5) - (t - 2)(2t). Notice that choice A correctly shows this subtraction pattern without expanding. A common error is forgetting to distribute the negative sign when subtracting (t - 2)(2t), which would lead to an incorrect positive term. When applying the quotient rule, maintain the exact form [g'h - gh']/h² to minimize errors.
In economics, $M(q)=\dfrac{\ln q}{q^2+3}$ . What is $M'(q)$ for $q>0$ ?
Answer: $\dfrac{\frac{1}{q}(q^2+3)-\ln q\,(2q)}{(q^2+3)^2}$ To differentiate M(q) = ln q/(q² + 3), we need d/dq(ln q) = 1/q and d/dq(q² + 3) = 2q. The quotient rule yields M'(q) = [(1/q)(q² + 3) - ln q·(2q)]/(q² + 3)². The first term in the numerator is (q² + 3)/q, and the second term is 2q ln q, giving us [(q² + 3)/q - 2q ln q]/(q² + 3)². A typical error is forgetting that ln q differentiates to 1/q, not just 1. Remember: when applying the quotient rule, carefully track each derivative and maintain the correct subtraction order in the numerator.
A temperature index is $T(x)=\dfrac{x^4}{x^3+1}$ . What is $T'(x)$ ?
Answer: $\dfrac{(4x^3)(x^3+1)-(x^4)(3x^2)}{(x^3+1)^2}$ 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.
If $g(x)=\dfrac{x^2+1}{\cos x}$ , what is $g'(x)$ ?
Answer: $\dfrac{2x\cos x-(x^2+1)(-\sin x)}{(\cos x)^2}$ For g(x) = (x² + 1)/cos x, we differentiate the numerator to get 2x and the denominator to get -sin x. The quotient rule gives g'(x) = [2x·cos x - (x² + 1)·(-sin x)]/(cos x)². This simplifies to [2x·cos x + (x² + 1)·sin x]/(cos x)² because subtracting a negative becomes addition. A common error is forgetting that the derivative of cos x is -sin x (with a negative sign), which affects the sign in the numerator. To avoid confusion, always write out the derivatives explicitly before substituting into the quotient rule formula.
For a population ratio $R(x)=\dfrac{e^x}{x^2+1}$ , what is $R'(x)$ ?
Answer: $\dfrac{e^x(x^2+1)-e^x(2x)}{(x^2+1)^2}$ For R(x) = $e^x$ /(x² + 1), we use the quotient rule with g(x) = $e^x$ (so g'(x) = $e^x$ ) and h(x) = x² + 1 (so h'(x) = 2x). The quotient rule formula gives R'(x) = [ $e^x$ (x² + 1) - $e^x$ (2x)]/(x² + 1)². Notice that the numerator follows the pattern g'h - gh', with a minus sign between the terms. A common mistake is adding these terms instead of subtracting, which would give the incorrect expression in choice A. The denominator must be [h(x)]² = (x² + 1)², not just (x² + 1). To avoid sign errors, remember that the quotient rule always has subtraction in the numerator: derivative of top times bottom minus top times derivative of bottom.
Let $h(x)=\dfrac{2x+5}{x^2}$ . What is $h'(x)$ for $x\ne0$ ?
Answer: $\dfrac{2x^2-(2x+5)(2x)}{x^4}$ 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.
A growth efficiency is $E(t)=\dfrac{\sqrt{t}}{t+2}$ for $t>0$ . What is $E'(t)$ ?
Answer: $\dfrac{\left(\frac{1}{2\sqrt{t}}\right)(t+2)-\sqrt{t}(1)}{(t+2)^2}$ The quotient rule states [g'(t) h(t) - g(t) h'(t)] / [h(t)]^2 for E(t) = g(t)/h(t). Here, g(t) = √t with g'(t) = 1/(2√t), h(t) = t + 2 with h'(t) = 1, giving [ (1/(2√t)) (t + 2) - √t (1) ] / (t + 2)², matching choice A. Common mistakes include wrong order of subtraction in numerator. Omitting the square on denominator results in choice C. Errors in square root derivatives are prevalent. Always compute g' and h' first and confirm with known rules before proceeding.
A cost index is $I(t)=\dfrac{\ln t}{t^2-1}$ . What is $I'(t)$ for $t>0$ ?
Answer: $\dfrac{\frac{1}{t}(t^2-1)-(\ln t)(2t)}{(t^2-1)^2}$ To differentiate I(t) = ln(t)/(t² - 1), identify the numerator g(t) = ln(t) with g'(t) = 1/t, and denominator h(t) = t² - 1 with h'(t) = 2t. Applying the quotient rule: I'(t) = [(1/t)(t² - 1) - ln(t)(2t)]/(t² - 1)². The key is maintaining the correct order: derivative of numerator times denominator minus numerator times derivative of denominator. A frequent error is forgetting to square the denominator, using just (t² - 1) instead of (t² - 1)². Another mistake would be switching the order of subtraction in the numerator. Remember the mnemonic 'low d-high minus high d-low, all over low squared' to keep the quotient rule structure correct.
A ratio is $p(x)=\dfrac{\sqrt{x}}{x+5}$ . What is $p'(x)$ for $x>0$ ?
Answer: $\dfrac{\frac{1}{2\sqrt{x}}(x+5)-\sqrt{x}}{(x+5)^2}$ To find p'(x) for p(x) = √x/(x + 5), we need the derivative of √x, which is 1/(2√x), and the derivative of x + 5, which is 1. Applying the quotient rule: p'(x) = [(1/2√x)(x + 5) - √x·1]/(x + 5)². The numerator simplifies to (x + 5)/(2√x) - √x, which we can rewrite with a common denominator as [(x + 5) - 2x]/(2√x) = (5 - x)/(2√x). A frequent mistake is incorrectly differentiating √x as 1/√x instead of 1/(2√x). Always double-check power rule applications when dealing with fractional exponents.
A demand ratio is $D(p)=\dfrac{p+4}{p^2-1}$ . What is $D'(p)$ ?
Answer: $\dfrac{(1)(p^2-1)-(p+4)(2p)}{(p^2-1)^2}$ 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.

Start this quiz