This temperature model requires applying the power rule to find the rate of change. The power rule states that for any term $at^n$, the derivative is n·at^(n-1). For T(t) = $t^8$ - $6t^4$ + $9t^2$ + 1, we differentiate each term: $t^8$ becomes $8t^7$, $-6t^4$ becomes $-24t^3$, $9t^2$ becomes 18t, and the constant 1 becomes 0. Therefore, T'(t) = $8t^7$ - $24t^3$ + 18t. Choice D incorrectly adds the constant 1 to the derivative, failing to recognize that constants differentiate to zero. Remember the power rule strategy: bring down the exponent as a multiplier, reduce the exponent by 1, and constants vanish.

This problem requires applying the power rule to find the derivative of a position function. The power rule states that if f(x) = $x^n$, then f'(x) = nx^(n-1). For s(t) = $4t^7$ - $3t^4$ + 6t - 9, we apply the rule term by term: the derivative of $4t^7$ is $28t^6$, the derivative of $-3t^4$ is $-12t^3$, the derivative of 6t is 6, and the derivative of the constant -9 is 0. Choice E incorrectly keeps the constant term -9, which is a common error since constants have zero derivatives. Remember: when using the power rule, multiply the coefficient by the exponent and reduce the exponent by 1.

Finding C'(x) requires applying the power rule to the cost function. The power rule states that the derivative of $x^n$ is nx^(n-1). For C(x) = $x^9$ - $4x^5$ + $2x^2$ - 1, we differentiate each term: $x^9$ becomes $9x^8$, $-4x^5$ becomes $-20x^4$, $2x^2$ becomes 4x, and the constant -1 becomes 0. Choice A incorrectly keeps the constant -1 in the derivative, which violates the rule that constants have zero derivatives. Always drop constant terms when applying the power rule to find derivatives.

The power rule is the key skill here, which states that the derivative of a term like a $x^n$ is a n $x^{n-1}$. To find R'(x) from R(x) = $x^7$ - $4x^5$ + $6x^2$, begin with the first term: the derivative of $x^7$ (which is $1x^7$) is 17 $x^6$ = $7x^6$. Then, the derivative of $-4x^5$ is -45 $x^4$ = $-20x^4$, and the derivative of $6x^2$ is 6*2 $x^1$ = 12x. Combining these gives $7x^6$ - $20x^4$ + 12x. A tempting distractor like choice D, $7x^6$ - $4x^4$ + 12x, fails by forgetting to multiply the -4 by the exponent 5. Remember, a transferable strategy for the power rule is to handle each term independently, multiply by the exponent, and subtract one from the exponent, ensuring constants vanish.

The power rule is the key skill here, which states that the derivative of a term like a $x^n$ is a n $x^{n-1}$, and constants have a derivative of zero. To find C'(x) from C(x) = $9x^4$ - $2x^3$ + x - 12, begin with the first term: the derivative of $9x^4$ is 94 $x^3$ = $36x^3$. Then, the derivative of $-2x^3$ is -23 $x^2$ = $-6x^2$, and the derivative of x (which is $1x^1$) is 1*1 $x^0$ = 1. The constant -12 differentiates to 0, resulting in $36x^3$ - $6x^2$ + 1. A tempting distractor like choice D, $36x^3$ - $2x^2$ + 1, fails by not multiplying the -2 coefficient by its exponent of 3. Remember, a transferable strategy for the power rule is to handle each term independently, multiply by the exponent, and subtract one from the exponent, ensuring constants vanish.

To find d'(x) for the displacement function d(x) = $-6x^5$ + $11x^2$ - 4x + 13, we apply the power rule to each term. Using the rule that the derivative of $x^n$ is nx^(n-1): the derivative of $-6x^5$ is $5·(-6)x^4$ = $-30x^4$, the derivative of $11x^2$ is $2·11x^1$ = 22x, the derivative of -4x is $-4·1x^0$ = -4, and the derivative of 13 is 0. Choice C $(-6x^4$ + 11x - 4) makes the error of only reducing exponents without multiplying by them, missing the key multiplication step in the power rule. Remember: multiply the coefficient by the exponent, then reduce the exponent by 1.

Finding R'(x) for the revenue function R(x) = $3x^7$ + $4x^2$ - x + 8 requires applying the power rule to each term. The power rule gives us: the derivative of $3x^7$ is $7·3x^6$ = $21x^6$, the derivative of $4x^2$ is $2·4x^1$ = 8x, the derivative of -x (which is $-x^1$) is $1·(-1)x^0$ = -1, and the derivative of 8 is 0. Choice C $(3x^6$ + 4x - 1) incorrectly applies the power rule by not multiplying the coefficients by the exponents for the first two terms. Remember that the power rule requires both multiplication by the exponent and reduction of the exponent by 1.

To find C'(x), we apply the power rule to each term of the cost function C(x) = $12x^4$ - $5x^3$ + 2x + 11. Using the power rule $d/dx(x^n$) = nx^(n-1), we get: the derivative of $12x^4$ is $4·12x^3$ = $48x^3$, the derivative of $-5x^3$ is $3·(-5)x^2$ = $-15x^2$, the derivative of 2x is $2·1x^0$ = 2, and the derivative of the constant 11 is 0. Choice C $(12x^3$ - $5x^2$ + 2) makes the common error of only bringing down the exponent without multiplying by the original coefficient. The key to the power rule is to multiply the coefficient by the exponent before reducing the exponent by 1.

This problem involves using the power rule to differentiate the polynomial $h(t) = 5t^4 - 3t^2 + 7t - 9$. To apply the power rule, take each term separately: for $5t^4$, the derivative is $54 t^3 = 20t^3$. Next, for $-3t^2$, it's $-32 t^1 = -6t$. Then, for $7t$, it's $7*1 t^0 = 7$, and the constant $-9$ differentiates to $0$. A common mistake is seen in choice C, where exponents aren't reduced, leading to incorrect terms like $20t^4$ instead of decreasing the powers. Remember, a key strategy for the power rule is to multiply each term's coefficient by its exponent and then subtract one from that exponent, ensuring constants disappear in the derivative.

This problem involves the power rule for T(t) = $3t^{-2}$ - $5t^4$ + 11, featuring a negative exponent. Differentiate $3t^{-2}$ to 3*(-2) $t^{-3}$ = $-6t^{-3}$. For $-5t^4$, it's -5*4 $t^3$ = $-20t^3$. The constant +11 becomes 0. Choice D is tempting but wrong as it omits the negative sign from the first term's differentiation. A general strategy for the power rule, even with negative or fractional exponents, is to multiply each coefficient by its exponent and subtract one from the exponent, dropping constants.