This problem requires applying the power rule to find the derivative of the polynomial q(x). The power rule states that for c $x^n$, the derivative is c n $x^{n-1}$, treating terms separately. Differentiating $8x^2$ gives 16x, and $+5x^7$ becomes $+35x^6$. Combining these yields q'(x) = 16x + $35x^6$. A tempting distractor is choice D, which incorrectly uses $35x^7$ instead of $35x^6$. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

This problem requires applying the power rule to find the derivative of a profit function given as a polynomial. According to the power rule, the derivative of c $t^n$ is c n $t^{n-1}$, applied to each term individually. For $6t^3$, it becomes $18t^2$; for -8t, it is -8; and the constant +1 differentiates to 0. Thus, P'(t) = $18t^2$ - 8. A tempting distractor is choice D, which incorrectly adds back the constant 1, but constants vanish in differentiation. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

This problem requires applying the power rule to find the derivative of a simple polynomial model. The power rule dictates that for c $t^n$, the derivative is c n $t^{n-1}$, handling each term separately. Differentiating $t^6$ $(1t^6$) gives $6t^5$, $+4t^2$ becomes +8t, and -11 turns to 0. Combining yields p'(t) = $6t^5$ + 8t. A tempting distractor is choice D, which incorrectly includes the constant -11 in the derivative. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

This problem requires applying the power rule to find the marginal revenue function. The power rule tells us that $d/dx[x^n$] = nx^(n-1). For R(x) = $x^7$ - $6x^4$ + 2x, we apply the rule to each term: the derivative of $x^7$ is $7x^6$, the derivative of $-6x^4$ is $-6·4x^3$ = $-24x^3$, and the derivative of 2x (which is $2x^1$) is $2·1x^0$ = 2. Therefore, R'(x) = $7x^6$ - $24x^3$ + 2. Choice C incorrectly computes the middle term as $-6x^3$, forgetting to multiply the coefficient by the power. Always remember the complete power rule formula: multiply the coefficient by the exponent, then reduce the exponent by one.

This problem involves finding the marginal profit by applying the power rule. The power rule tells us that $d/dx[x^n$] = nx^(n-1). For P(x) = $6x^3$ - $11x^2$ + 8x - 1, we differentiate term by term: the derivative of $6x^3$ is $6·3x^2$ = $18x^2$, the derivative of $-11x^2$ is -11·2x = -22x, the derivative of 8x is 8, and the derivative of the constant -1 is 0. Thus, P'(x) = $18x^2$ - 22x + 8. Choice D incorrectly computes the middle term as -11x, forgetting to multiply by the power 2. Always apply the complete power rule: new coefficient = old coefficient × old exponent.

This problem requires applying the power rule to differentiate m(x) = $x^10$ - $4x^7$ + 8x. The power rule states that $d/dx[x^n$] = nx^(n-1). Applying this to each term: the derivative of $x^10$ is $10x^9$, the derivative of $-4x^7$ is $7·(-4)x^6$ = $-28x^6$, and the derivative of 8x is $8·1x^0$ = 8. Thus, m'(x) = $10x^9$ - $28x^6$ + 8. Choice E incorrectly omits the derivative of the 8x term, forgetting that linear terms have constant derivatives. Always differentiate every term in the function, including linear terms whose derivatives are constants.

To find C'(x), we must apply the power rule to each term of the cost function. The power rule gives us $d/dx[x^n$] = nx^(n-1). For C(x) = $12x^2$ - $4x^6$ + $x^3$, we get: $12x^2$ becomes $2·12x^1$ = 24x, $-4x^6$ becomes $6·(-4)x^5$ = $-24x^5$, and $x^3$ becomes $3·x^2$ = $3x^2$. Choice B (24x - $4x^5$ + $3x^2$) incorrectly differentiates $-4x^6$ as $-4x^5$, failing to multiply by the exponent 6. The systematic approach is to identify each term's coefficient and exponent, multiply them together, then reduce the exponent by 1.

Finding C'(n) requires applying the power rule to each term of the cost function. The power rule states that $d/dx[x^n$] = nx^(n-1). For C(n) = $9n^3$ + $2n^2$ - n + 6, we differentiate term by term: $9n^3$ becomes $9·3n^2$ = $27n^2$, $2n^2$ becomes $2·2n^1$ = 4n, -n becomes $-1·1n^0$ = -1, and the constant 6 becomes 0. Therefore, C'(n) = $27n^2$ + 4n - 1. Choice C shows the error of forgetting to multiply by the original exponent, giving $9n^2$ instead of $27n^2$. When using the power rule, always remember to multiply the coefficient by the exponent before reducing the exponent.

This problem requires applying the power rule to find the derivative of a polynomial function. The power rule states that if f(x) = $x^n$, then f'(x) = nx^(n-1). For h(t) = $5t^4$ - $2t^3$ + t, we apply the power rule to each term: the derivative of $5t^4$ is $5·4t^3$ = $20t^3$, the derivative of $-2t^3$ is $-2·3t^2$ = $-6t^2$, and the derivative of t (which is $t^1$) is $1·t^0$ = 1. Therefore, h'(t) = $20t^3$ - $6t^2$ + 1. A common error would be to forget to multiply by the original coefficient, giving $4t^3$ - $3t^2$ + 1 instead. When differentiating polynomials, always remember to multiply the power by the coefficient before reducing the exponent by one.

This problem requires applying the power rule to find the derivative of the polynomial f(x). According to the power rule, the derivative of c $x^n$ is c n $x^{n-1}$, applied term by term. For $10x^5$, it becomes $50x^4$; for $-3x^4$, it is $-12x^3$; and -2x differentiates to -2. Thus, f'(x) = $50x^4$ - $12x^3$ - 2. A tempting distractor is choice D, which wrongly changes -2 to -2x. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.