This problem tests the basic derivative rules for constants, sums, and differences. To find d'(x) for d(x) = $12x^3$ + $7x^2$ - 4x + 1, apply the power rule to the first term: the derivative of $12x^3$ is $123x^{2}$ = $36x^2$. For the second term, the derivative of $+7x^2$ is $72x^{1}$ = 14x, and for -4x it is $-41x^{0}$ = -4. The constant term +1 has a derivative of 0, so it drops out. A tempting distractor is choice B, which adds +1, but constants vanish in derivatives. Always differentiate term by term, applying the power rule to each polynomial term and setting the derivative of constants to zero.

This problem tests the basic derivative rules for constants, sums, and differences. To find s'(x) for $s(x) = -5x^3 + 4x - 9$, apply the power rule to the first term: the derivative of $-5x^3$ is $-53x^{2} = -15x^2$. For the second term, the derivative of $+4x$ is $41x^{0} = 4$. The constant term $-9$ has a derivative of 0, so it is omitted from the result. A tempting distractor is choice B, which subtracts the constant $-9$ again, but constants do not persist in derivatives. Always differentiate term by term, applying the power rule to each polynomial term and setting the derivative of constants to zero.

This problem tests the basic derivative rules for constants, sums, and differences. To find h'(t) for h(t) = 9 - $2t^5$ + $6t^2$, apply the power rule to the second term: the derivative of $-2t^5$ is $-25t^{4}$ = $-10t^4$. For the third term, the derivative of $+6t^2$ is $62t^{1}$ = 12t. The constant term 9 has a derivative of 0, so it is not included. A tempting distractor is choice C, which adds +9, but constants disappear upon differentiation. Always differentiate term by term, applying the power rule to each polynomial term and setting the derivative of constants to zero.

This problem tests the basic derivative rules for constants, sums, and differences. To find V'(t) for V(t) = $7t^4$ - $3t^2$ + 12, apply the power rule to the first term: the derivative of $7t^4$ is $74t^{3}$ = $28t^3$. For the second term, the derivative of $-3t^2$ is $-32t^{1}$ = -6t. The constant term +12 has a derivative of 0, so it disappears in the derivative. A tempting distractor is choice A, which includes +12, but this is incorrect because the derivative of a constant is always zero. Always differentiate term by term, applying the power rule to each polynomial term and setting the derivative of constants to zero.

This problem requires finding the marginal revenue using basic derivative rules. We differentiate each term separately: the derivative of 3x⁶ is 18x⁵ (3·6·x⁵), the derivative of -11x³ is -33x² (-11·3·x²), and the derivative of 4x is 4. Combining these gives R'(x) = 18x⁵ - 33x² + 4. Choice B incorrectly writes the last term as 4x instead of 4, failing to recognize that d/dx(4x) = 4, not 4x. The systematic approach is to apply the power rule to each term, remembering that linear terms become constants.

This problem asks us to find the velocity function by applying basic derivative rules to the position function. We differentiate term by term: the derivative of 5x³ is 15x² (multiply by the exponent 3 and reduce the power by 1), the derivative of 2x is 2 (since x¹ becomes 1·x⁰ = 1), and the derivative of the constant -9 is 0. Therefore, s'(x) = 15x² + 2 + 0 = 15x² + 2. Choice E incorrectly keeps the x in the second term (2x instead of 2), forgetting that d/dx(2x) = 2, not 2x. The key strategy is to handle each term independently using the power rule.

This problem requires applying basic derivative rules to find the rate of change of water volume. To find V'(t), we differentiate each term separately: the derivative of 7t⁴ is 28t³ (using the power rule: bring down the 4 and reduce the exponent by 1), the derivative of -3t² is -6t, and the derivative of the constant 12 is 0. Combining these results gives V'(t) = 28t³ - 6t + 0 = 28t³ - 6t. A common error would be to include the constant term in the derivative (choice A), but constants always have a derivative of zero. Remember: when differentiating a polynomial, apply the power rule to each term and constants disappear.

This problem requires finding f'(x) using basic derivative rules on a polynomial function. We differentiate each term: the derivative of -9x⁴ is -36x³ (multiply -9 by 4 and reduce the exponent), the derivative of 7x³ is 21x² (7·3·x²), and the derivative of the constant -5 is 0. Thus, f'(x) = -36x³ + 21x² + 0 = -36x³ + 21x². Choice B incorrectly includes the constant -5 in the derivative, but constants always have zero derivative. The key is to apply the power rule term by term, remembering that constant terms disappear.

This problem assesses your understanding of the basic derivative rules for constants, sums, and differences in polynomials. To compute T'(t), differentiate each term separately using the power rule. The derivative of $-7t^3$ is -7 times $3t^{2}$, which is $-21t^2$. The derivative of $+2t^2$ is 2 times 2t, which is +4t, and the derivative of +12 is 0, so they combine to $-21t^2$ + 4t. A tempting distractor like choice C changes +4t to -4t, but this fails because the positive sign remains for the quadratic term's derivative. Always remember to apply the power rule to each term independently and set the derivative of constants to zero.

This problem assesses your understanding of the basic derivative rules for constants, sums, and differences in polynomials. To compute h'(t), differentiate each term separately using the power rule. The derivative of $5t^4$ is 5 times $4t^{3}$, which is $20t^3$. The derivative of $-3t^2$ is -3 times 2t, which is -6t, and the derivative of +9 is 0, so they combine to $20t^3$ - 6t. A tempting distractor like choice D includes the +9, but this fails because the derivative of a constant is always zero. Always remember to apply the power rule to each term independently and set the derivative of constants to zero.