Calculating Higher-Order Derivatives - AP Calculus BC

1. The revenue is $R(x)=x^6-5x^3+4x$. What is the fourth derivative, $R^{(4)}(x)$?
   Answer: $360x^2$ This problem tests the ability to find the fourth derivative of a revenue function containing various powers of x. Beginning with R(x) = $x^6$ - $5x^3$ + 4x, we find R'(x) = $6x^5$ - $15x^2$ + 4, then R''(x) = $30x^4$ - 30x, followed by $R^{3}$(x) = $120x^3$ - 30. The fourth derivative is $R^{4}$(x) = $360x^2$. Choice C incorrectly retains the constant -30 from the third derivative, not recognizing that constants vanish when differentiated. For higher-order derivatives of polynomials, track how each term's power decreases until terms with initial power less than the derivative order disappear entirely.
2. A particle's position is $s(t)=3t^5-2t^4+7t^2-9t$. What is $s^{(3)}(t)$?
   Answer: $180t^2-48t$ This problem requires calculating the third derivative of a polynomial position function, demonstrating higher-order differentiation skills. Starting with s(t) = $3t^5$ - $2t^4$ + $7t^2$ - 9t, the first derivative is s'(t) = $15t^4$ - $8t^3$ + 14t - 9. The second derivative is s''(t) = $60t^3$ - $24t^2$ + 14. The third derivative is $s^{3}$(t) = $180t^2$ - 48t. Choice A incorrectly includes the constant 14 from the second derivative, failing to recognize that the derivative of a constant is zero. When finding higher-order derivatives, systematically apply the power rule at each stage, remembering that constants disappear and the power decreases by one each time.
3. A profit function is $P(x)=7x^3-9x^2+4x+1$. What is $P^{(3)}(x)$?
   Answer: $42$ This problem requires finding the third derivative of a profit function, testing proficiency in calculating higher-order derivatives. Given P(x) = $7x^3$ - $9x^2$ + 4x + 1, we compute P'(x) = $21x^2$ - 18x + 4, P''(x) = 42x - 18, and $P^{3}$(x) = 42. A common error would be to write 42x, forgetting that we're differentiating 42x to get the constant 42. When finding the nth derivative of a polynomial, remember that $x^n$ differentiates to n! (a constant), and all terms of degree less than n become zero.
4. For $g(x)=x^4-6x^2+5$, what is the third derivative $g^{(3)}(x)$?
   Answer: $24x$ This problem requires calculating higher-order derivatives of a polynomial function. Begin with g(x) = $x^4$ - $6x^2$ + 5 and compute the first derivative: g'(x) = $4x^3$ - 12x. The second derivative is g''(x) = $12x^2$ - 12, and the third is g'''(x) = 24x. Each step applies the power rule, eliminating constant terms and reducing degrees progressively. A tempting distractor like 24x - 12 could result from forgetting to differentiate the constant term in the second derivative. A transferable strategy for higher-order derivatives of polynomials is to apply the power rule repeatedly, tracking the factorial-like coefficient reductions until the desired order is reached.
5. A model uses $f(t)=3t^5-4t^3+2t$. What is the fourth derivative $f^{(4)}(t)$?
   Answer: $360t$ This problem requires calculating higher-order derivatives of a polynomial function. Start with f(t) = $3t^5$ - $4t^3$ + 2t and find the first derivative: f'(t) = $15t^4$ - $12t^2$ + 2. The second derivative is f''(t) = $60t^3$ - 24t, the third is f'''(t) = $180t^2$ - 24, and the fourth is $f^{(4)}$(t) = 360t. Notice how each differentiation reduces the degree of each term by one and multiplies the coefficient by the original exponent, which decreases accordingly. A tempting distractor like $360t^2$ might arise if someone mistakenly stops at the third derivative and adds an extra t factor. A transferable strategy for higher-order derivatives of polynomials is to apply the power rule repeatedly, tracking the factorial-like coefficient reductions until the desired order is reached.
6. A height function is $h(x)=7x^5-10x^2+6x-1$. What is $h^{(4)}(x)$?
   Answer: $840x$ Finding the fourth derivative of a height function demonstrates how polynomial terms of different degrees behave under repeated differentiation. Starting with h(x) = $7x^5$ - $10x^2$ + 6x - 1, successive derivatives yield h'(x) = $35x^4$ - 20x + 6, h''(x) = $140x^3$ - 20, $h^{3}$(x) = $420x^2$. The fourth derivative is $h^{4}$(x) = 840x. Choice D incorrectly gives 840 as a constant, missing that we're differentiating $420x^2$, not a constant term. For efficient higher-order differentiation, identify which terms will survive to the desired derivative order before beginning calculations.
7. A population model is $P(t)=4t^6-3t^4+2t^3-5$. What is $P^{(3)}(t)$?
   Answer: $480t^3-72t+12$ This problem tests third derivative calculation for a population model with multiple polynomial terms. From P(t) = $4t^6$ - $3t^4$ + $2t^3$ - 5, we get P'(t) = $24t^5$ - $12t^3$ + $6t^2$, then P''(t) = $120t^4$ - $36t^2$ + 12t. The third derivative is $P^{3}$(t) = $480t^3$ - 72t + 12. Choice B incorrectly shows the constant as 6 instead of 12, likely from an error when differentiating 12t in the second derivative. When computing higher derivatives, maintain precision with coefficients and remember that the derivative of kt is k, not k/2.
8. For the revenue model $R(x)=2x^5+x^3-4x^2+9$, compute the second derivative $R''(x)$.
   Answer: $40x^3+6x-8$ This problem requires calculating the second derivative of a revenue function, testing the skill of higher-order differentiation. Starting with R(x) = $2x^5$ + $x^3$ - $4x^2$ + 9, we find R'(x) = $10x^4$ + $3x^2$ - 8x, then R''(x) = $40x^3$ + 6x - 8. Students might incorrectly write $40x^3$ + 6x + 8, confusing the sign when differentiating -8x, but the derivative of -8x is -8, not +8. For accurate higher-order derivatives, maintain careful attention to signs and coefficients throughout each differentiation step.
9. A particle's position is $s(t)=3t^5-4t^3+2t$. What is the fourth derivative $s^{(4)}(t)$?
   Answer: $360t$ Higher-order differentiation involves repeatedly applying the derivative operation to find rates of change beyond velocity and acceleration, such as jerk or snap in this position function. Start with s(t) = $3t^5$ - $4t^3$ + 2t; the first derivative is s'(t) = $15t^4$ - $12t^2$ + 2. The second derivative is s''(t) = $60t^3$ - 24t, and the third is s'''(t) = $180t^2$ - 24. Finally, the fourth derivative is $s^{(4)}$(t) = 360t, as the constant term vanishes. A tempting distractor like 180t might arise from stopping at the third derivative and halving the coefficient incorrectly. A transferable strategy for higher-order derivatives of polynomials is to recognize that each differentiation reduces the degree by one, eventually yielding zero for degrees less than the order.
10. For $g(x)=x^8-4x^6+2x^2$, determine the sixth derivative $g^{(6)}(x)$.
    Answer: $20160x^2-2880$ Computing the sixth derivative of a high-degree polynomial demonstrates advanced skill in calculating higher-order derivatives. For g(x) = $x^8$ - $4x^6$ + $2x^2$, we note that the $x^2$ term vanishes after two differentiations, leaving us to track $x^8$ and $x^6$ terms: $g^{6}$(x) = (8×7×6×5×4×3)$x^2$ - 4(6×5×4×3×2×1) = $20160x^2$ - 2880. Students might incorrectly compute the factorial products, perhaps getting $6720x^2$ for the first term by missing a factor. For high-order derivatives, use the formula that the nth derivative of $x^k$ is k!/(k-n)! × $x^{k-n}$ when k ≥ n, and zero otherwise.
11. A signal is $g(x)=e^{3x}$. What is the fifth derivative $g^{(5)}(x)$?
    Answer: $243e^{3x}$ Higher-order differentiation of exponential functions highlights their unique property of retaining the same form scaled by constants, useful in signal processing. For g(x) = $e^{3x}$, each derivative multiplies by 3, so the first is $3e^{3x}$ and the second is $9e^{3x}$. The third is $27e^{3x}$, the fourth is $81e^{3x}$, and the fifth is $243e^{3x}$. This pattern arises because the derivative of $e^{kx}$ is k times itself. A tempting distractor like $81e^{3x}$ might come from miscounting the number of differentiations to the fourth instead of fifth. Generally, for $e^{kx}$, the nth derivative is $k^n$ $e^{kx}$, providing a quick formula for any order.
12. In a model, $R(x)=x^4-6x^2+5$. What is the fourth derivative, $R^{(4)}(x)$?
    Answer: $24$ This problem asks for the fourth derivative of a polynomial, requiring four successive differentiations. Starting with $R(x)=x^4-6x^2+5$, we get $R'(x)=4x^3-12x$, then $R''(x)=12x^2-12$, then $R'''(x)=24x$, and finally $R^{(4)}(x)=24$. A tempting error is to think the fourth derivative of $x^4$ is 0 (choice B), but we must remember that $x^4$ differentiates to $4x^3$, then $12x^2$, then $24x$, then $24$. For polynomial derivatives, the $n$th derivative of $x^n$ is $n!$, while higher powers of $x$ still have non-zero derivatives.
13. A wave is $w(x)=\sin(4x)$. What is the third derivative $w^{(3)}(x)$?
    Answer: $-64\cos(4x)$ Higher-order differentiation of sine waves reveals cyclic patterns with amplitude growth, essential in wave analysis. For w(x) = sin(4x), the first derivative is 4 cos(4x). The second is -16 sin(4x), and the third is -64 cos(4x), as each step multiplies by 4 and alternates the trig function. The sign and function switch predictably. A tempting distractor like 64 cos(4x) could overlook the negative sign from the second derivative carrying over. For sin(kx), the nth derivative is $k^n$ sin(kx + nπ/2), aiding in tracking phase and magnitude for higher orders.
14. Given $F(t)=t^8-2t^6+3t^2$, what is the seventh derivative $F^{(7)}(t)$?
    Answer: $40320t$ This problem requires calculating higher-order derivatives of a polynomial function. For F(t) = $t^8$ - $2t^6$ + $3t^2$, the seventh derivative is 40320t, stemming from the $t^8$ term. The coefficient is 8!/1! = 40320, with lower terms yielding zero at this order. Compute progressively: each derivative reduces degrees and multiplies by descending exponents. A tempting distractor like 5040t might come from using 7! instead of 8! for the coefficient. A transferable strategy for higher-order derivatives of polynomials is to apply the power rule repeatedly, tracking the factorial-like coefficient reductions until the desired order is reached.
15. The cost function is $C(x)=7x^4-5x^2+9$. What is the third derivative $C^{(3)}(x)$?
    Answer: $168x$ This problem tests your ability to find higher-order derivatives, specifically the third derivative of a cost function. Given C(x) = $7x^4$ - $5x^2$ + 9, we differentiate step by step: $C'(x) = 28x^3 - 10x$, $C''(x) = 84x^2 - 10$, and $C'''(x) = 168x$. Choice B ($84x^2 - 10$) might seem correct if you stop at the second derivative, a common mistake when tracking multiple differentiations. For polynomial functions, the nth derivative eliminates all terms of degree less than n, which is why the constant and $x^2$ terms vanish by the third derivative.
16. For $p(x)=x^4\ln(x)$ with $x>0$, what is the second derivative $p''(x)$?
    Answer: $12x^2\ln(x)+7x^2$ Higher-order differentiation of products like $x^4$ ln(x) requires repeated application of the product rule, revealing logarithmic growth patterns. For p(x) = $x^4$ ln(x), the first derivative is $4x^3$ ln(x) + $x^3$ using the product rule. The second derivative applies the product rule twice: differentiating to get $12x^2$ ln(x) + $7x^2$. This combines terms from both parts of the product. A tempting distractor like $12x^2$ ln(x) + $8x^2$ might occur from an arithmetic error in combining constants. A useful strategy is to factor out common powers after each differentiation to simplify higher-order computations.
17. A function is $r(t)=6t^4+t^3-8t$. What is the third derivative $r^{(3)}(t)$?
    Answer: $144t+6$ This problem requires calculating higher-order derivatives of a polynomial function. With r(t) = $6t^4$ + $t^3$ - 8t, the first derivative is $24t^3$ + $3t^2$ - 8. The second is $72t^2$ + 6t, and the third is 144t + 6. Constants and lower terms vanish appropriately with each step. A tempting distractor like 144t could occur if the constant from the second derivative's linear term is ignored. A transferable strategy for higher-order derivatives of polynomials is to apply the power rule repeatedly, tracking the factorial-like coefficient reductions until the desired order is reached.
18. Given $g(x)=e^{2x}$, what is the third derivative $g^{(3)}(x)$?
    Answer: $8e^{2x}$ This problem tests calculating higher-order derivatives of exponential functions with chain rule applications. For g(x) = $e^{2x}$, the first derivative is g'(x) = $2e^{2x}$ using the chain rule. The second derivative is g''(x) = 2·$2e^{2x}$ = $4e^{2x}$, and the third derivative is g'''(x) = 2·$4e^{2x}$ = $8e^{2x}$. Choice C ($4e^{2x}$) might attract students who forget to multiply by the chain rule factor of 2 at each differentiation step. For exponential functions $e^{kx}$, each derivative multiplies by the constant k, so the nth derivative is $k^n$·$e^{kx}$.
19. A revenue model is $p(t)=9t^4-2t^2+t$. What is the third derivative $p^{(3)}(t)$?
    Answer: $216t$ This problem asks for the third derivative, requiring three successive applications of the power rule. From $p(t)=9t^4-2t^2+t$, we get $p'(t)=36t^3-4t+1$, then $p''(t)=108t^2-4$, and finally $p^{(3)}(t)=216t$. Students might incorrectly include a constant term like $216t-4$ (choice D), not realizing that the constant $-4$ in the second derivative becomes 0 in the third derivative. Remember that the derivative of any constant is 0, so constant terms disappear with each differentiation.
20. Let $m(x)=x^6-5x^3+4x^2$. What is the fourth derivative $m^{(4)}(x)$?
    Answer: $360x^2$ This problem requires calculating higher-order derivatives of a polynomial function. For m(x) = $x^6$ - $5x^3$ + $4x^2$, the first derivative is $6x^5$ - $15x^2$ + 8x. The second is $30x^4$ - 30x + 8, the third is $120x^3$ - 30, and the fourth is $360x^2$. Terms of degree less than the derivative order become zero. A tempting distractor like $120x^3$ - 30 might be selected if someone computes only up to the third derivative. A transferable strategy for higher-order derivatives of polynomials is to apply the power rule repeatedly, tracking the factorial-like coefficient reductions until the desired order is reached.

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