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AP Calculus BC Quiz
Practice Calculating Higher Order Derivatives in AP Calculus BC with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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A signal is modeled by g(t)=t8−2t5+3t2. What is g(6)(t)?
This quiz focuses on Calculating Higher Order Derivatives, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus BC.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
A signal is modeled by g(t)=t8−2t5+3t2. What is g(6)(t)?
Explanation: Finding the sixth derivative of a high-degree polynomial requires careful tracking of how terms evolve through multiple differentiations. Starting with g(t) = t^8 - 2t^5 + 3t^2, the t^8 term contributes 8!/(8-6)! × t^2 = 8×7×6×5×4×3 × t^2 = 20160t^2 to the sixth derivative, while the t^5 and t^2 terms vanish since their degrees are less than 6. Therefore, g^(6)(t) = 20160t^2. Choice B shows 6720t^2, which appears to be 20160/3, suggesting a calculation error. For nth derivatives of x^k where k ≥ n, use the formula k!/(k-n)! × x^(k-n) to efficiently compute the result.
For p(x)=x4ln(x) with x>0, what is the second derivative p′′(x)?
Explanation: 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.
A motion model uses v(t)=2t4−8t2+5. What is the second derivative v′′(t)?
Explanation: This problem requires finding the second derivative of a polynomial motion model. Given v(t) = 2t^4 - 8t^2 + 5, we differentiate: v'(t) = 8t^3 - 16t, and v''(t) = 24t^2 - 16. Choice E (24t - 16) incorrectly reduces the power of t^2 to t^1, perhaps confusing the second derivative with a third derivative. When finding second derivatives, remember that each differentiation reduces the power by exactly one, so t^3 becomes t^2, not t.
The cost function is C(x)=7x4−5x2+9. What is the third derivative C(3)(x)?
Explanation: 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)=28x3−10x, C′′(x)=84x2−10, and C′′′(x)=168x. Choice B (84x2−10) 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.
A model uses s(t)=3t5−4t3+2t for position. What is the fourth derivative s(4)(t)?
Explanation: This problem requires calculating higher-order derivatives, specifically the fourth derivative of a polynomial position function. Starting with s(t) = 3t^5 - 4t^3 + 2t, we differentiate: s'(t) = 15t^4 - 12t^2 + 2, s''(t) = 60t^3 - 24t, s'''(t) = 180t^2 - 24, and finally s^(4)(t) = 360t. Choice D (180t^2 - 24) is tempting because it's the third derivative, showing a common error of stopping one step too early. When finding higher-order derivatives of polynomials, remember that each differentiation reduces the degree by one, and the nth derivative of t^n is n!.
A model uses f(t)=3t5−4t3+2t. What is the fourth derivative f(4)(t)?
Explanation: 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.
The temperature model is T(x)=x6−2x3+7x. Find the fourth derivative, T(4)(x).
Explanation: This problem tests the ability to compute the fourth derivative of a polynomial, a key skill in calculating higher-order derivatives. Starting with T(x) = x^6 - 2x^3 + 7x, we differentiate: T'(x) = 6x^5 - 6x^2 + 7, T''(x) = 30x^4 - 12x, T'''(x) = 120x^3 - 12, and T^(4)(x) = 360x^2. Students might mistakenly think the -12 from the third derivative carries forward, yielding 360x^2 - 24, but constants vanish upon differentiation. For higher-order derivatives, track each term separately and remember that terms of degree less than the derivative order will eventually become zero.
A trajectory is y(x)=2x5−5x2+1. What is the fifth derivative y(5)(x)?
Explanation: This problem requires calculating higher-order derivatives of a polynomial function. Given y(x)=2x5−5x2+1, the fifth derivative is 240, a constant from the x5 term alone. Step-by-step: first is 10x4−10x, second is 40x3−10, third is 120x2, fourth is 240x, fifth is 240. Lower terms vanish before the fifth order. A tempting distractor like 0 could arise from mistakenly thinking all terms disappear by the fifth 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.
A particle’s position is s(t)=3t5−2t4+7t2−9t. What is s(3)(t)?
Explanation: 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.
A cost function is C(x)=9x5+4x4−7x3+2. What is C(4)(x)?
Explanation: Computing the fourth derivative of a cost function demonstrates systematic higher-order differentiation. From C(x) = 9x^5 + 4x^4 - 7x^3 + 2, we find C'(x) = 45x^4 + 16x^3 - 21x^2, then C''(x) = 180x^3 + 48x^2 - 42x, followed by C^(3)(x) = 540x^2 + 96x - 42. The fourth derivative is C^(4)(x) = 1080x + 96. Choice E incorrectly omits the constant 96, which comes from differentiating 96x in the third derivative. When finding higher derivatives, remember that linear terms become constants and constants vanish, so track all terms carefully through each differentiation step.
A polynomial signal is s(t)=t5+2t4−t2. What is s(3)(t)?
Explanation: This problem requires calculating higher-order derivatives of a polynomial function. Given s(t) = t^5 + 2t^4 - t^2, the first derivative is s'(t) = 5t^4 + 8t^3 - 2t. The second is s''(t) = 20t^3 + 24t^2 - 2, and the third is s'''(t) = 60t^2 + 48t. Each differentiation lowers the degrees and adjusts coefficients via the power rule. A tempting distractor like 60t^2 + 48t - 2 might arise from failing 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.
A quantity is modeled by h(t)=2t6−t4+7. What is h(5)(t)?
Explanation: This problem requires calculating higher-order derivatives of a polynomial function. Start with h(t) = 2t^6 - t^4 + 7 and find the first derivative: h'(t) = 12t^5 - 4t^3. The second is h''(t) = 60t^4 - 12t^2, the third is h'''(t) = 240t^3 - 24t, the fourth is h^{(4)}(t) = 720t^2 - 24, and the fifth is h^{(5)}(t) = 1440t. Constants vanish after the first derivative, and lower-degree terms disappear in higher orders. A tempting distractor like 1440t - 24 might occur if the constant from the fourth derivative is not differentiated away. 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.
A cost function is C(x)=7x3−5x+9. What is C(2)(x)?
Explanation: Finding the second derivative means differentiating twice, a key skill in analyzing concavity and acceleration. From C(x)=7x3−5x+9, the first derivative is C′(x)=21x2−5. The second derivative is C′′(x)=42x. Students might incorrectly get 21x (choice B) by forgetting to multiply by 2 when differentiating 21x2. Remember that each differentiation applies the power rule independently: multiply by the current exponent and reduce the exponent by 1.
For g(x)=x4−6x2+5, what is the third derivative g(3)(x)?
Explanation: 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.
A population model is P(t)=4t6−3t4+2t3−5. What is P(3)(t)?
Explanation: 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.
The revenue is R(x)=x6−5x3+4x. What is the fourth derivative, R(4)(x)?
Explanation: 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.
A utility function is U(x)=6x7+x5−8x4+3x. What is U(3)(x)?
Explanation: Computing the third derivative of a utility function with multiple terms tests systematic differentiation skills. From U(x) = 6x^7 + x^5 - 8x^4 + 3x, we find U'(x) = 42x^6 + 5x^4 - 32x^3 + 3, then U''(x) = 252x^5 + 20x^3 - 96x^2. The third derivative is U^(3)(x) = 1260x^4 + 60x^2 - 192x. Choice E incorrectly shows -48x instead of -192x, suggesting an error when differentiating -96x^2. For polynomial higher-order derivatives, verify each coefficient by checking that it equals the product of the previous coefficient and the power being reduced.
A utility function is U(x)=6x5−3x+1. What is the fourth derivative U(4)(x)?
Explanation: This problem involves calculating the fourth derivative of a polynomial utility function. Starting with U(x) = 6x^5 - 3x + 1, we differentiate: U'(x) = 30x^4 - 3, U''(x) = 120x^3, U'''(x) = 360x^2, and U^(4)(x) = 720x. Choice B (360x) represents the coefficient from the third derivative without the final differentiation, a common error when losing track of which derivative you're on. For x^n, the kth derivative has coefficient n!/(n-k)!, so for x^5 taken four times: 5×4×3×2 = 120×6 = 720.
A cost model is C(x)=x1. What is the fifth derivative C(5)(x)?
Explanation: Higher-order differentiation of reciprocal functions like this cost model produces alternating signs and factorial-like coefficients in the numerators. For C(x)=1/x=x−1, the first derivative is −1/x2. The second is 2/x3, the third is −6/x4, the fourth is 24/x5, and the fifth is −120/x6. Each differentiation multiplies by the negative of the current power. A tempting distractor like 120/x6 could forget the alternating negative sign for odd orders. A general strategy is to use the formula for the nth derivative of x−1: (−1)nn!/xn+1 for quick verification.
Let m(x)=x6−5x3+4x2. What is the fourth derivative m(4)(x)?
Explanation: 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.