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.

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.

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.

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.

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!.

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.

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.

This problem requires calculating higher-order derivatives of a polynomial function. Given $y(x) = 2x^5 - 5x^2 + 1$, the fifth derivative is $240$, a constant from the $x^5$ term alone. Step-by-step: first is $10x^4 - 10x$, second is $40x^3 - 10$, third is $120x^2$, 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.

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.

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.