This sensor output function requires applying the power rule to find its derivative. The power rule states that for any term $ax^n$, the derivative is n·ax^(n-1). For f(x) = $x^7$ - $4x^4$ + $2x^2$ - 11, we differentiate each term: $x^7$ becomes $7x^6$, $-4x^4$ becomes $-16x^3$, $2x^2$ becomes 4x, and the constant -11 becomes 0. Therefore, f'(x) = $7x^6$ - $16x^3$ + 4x. Choice D incorrectly includes the original constant -11 in the derivative, not recognizing that constants differentiate to zero. The power rule strategy is straightforward: for each term, multiply the coefficient by the exponent and subtract 1 from the exponent.

This problem asks us to find the derivative of a revenue function using the power rule. The power rule tells us that the derivative of $x^n$ is nx^(n-1). For R(x) = $9x^4$ - $2x^3$ + 6, we differentiate term by term: the derivative of $9x^4$ is $4·9x^3$ = $36x^3$, the derivative of $-2x^3$ is $3·(-2)x^2$ = $-6x^2$, and the derivative of the constant 6 is 0. Thus, R'(x) = $36x^3$ - $6x^2$. Choice D incorrectly adds the constant 6 to the derivative, failing to recognize that constants have zero derivatives. The key strategy with the power rule is to process each term independently: multiply by the exponent, then decrease the exponent by 1.

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 polynomial position function. The power rule states that if f(x) = $x^n$, then f'(x) = nx^(n-1). For s(t) = $4t^5$ - $3t^2$ + 7t - 9, we apply the power rule to each term: the derivative of $4t^5$ is $5·4t^4$ = $20t^4$, the derivative of $-3t^2$ is $2·(-3)t^1$ = -6t, the derivative of 7t is $7·1t^0$ = 7, and the derivative of the constant -9 is 0. Therefore, s'(t) = $20t^4$ - 6t + 7. Choice E incorrectly keeps the linear term 7t in addition to its derivative 7, showing a common error of not replacing terms with their derivatives. Remember: when applying the power rule, multiply the coefficient by the exponent and reduce the exponent by 1 for each term.

To find m'(x), we apply the power rule to each term of the polynomial. The power rule gives us that the derivative of $ax^n$ is nax^(n-1). For m(x) = $3x^10$ - $x^5$ + $4x^3$ + 9, we differentiate term by term: $3x^10$ becomes $30x^9$, $-x^5$ becomes $-5x^4$, $4x^3$ becomes $12x^2$, and the constant 9 becomes 0. Choice E incorrectly includes the constant 9 in the derivative, which is wrong since constants differentiate to zero. Remember the power rule process: multiply coefficient by exponent, reduce exponent by 1, and eliminate constants.

To find f'(x), we apply the power rule to each term of the polynomial. The power rule tells us that $d/dx[ax^n$] = nax^(n-1). For f(x) = $-6x^7$ + $3x^4$ - $x^3$ + 10, we differentiate term by term: $-6x^7$ becomes $-42x^6$, $3x^4$ becomes $12x^3$, $-x^3$ becomes $-3x^2$, and the constant 10 becomes 0. Choice B incorrectly includes the constant 10, failing to recognize that constants differentiate to zero. Remember the power rule pattern: bring down the exponent as a multiplier, then subtract 1 from the exponent.

To find h'(t), we apply the power rule to each term in the height function. The power rule gives us that if f(x) = $ax^n$, then f'(x) = nax^(n-1). For h(t) = $2t^8$ - $7t^2$ + 3, we differentiate term by term: $2t^8$ becomes $16t^7$, $-7t^2$ becomes -14t, and the constant 3 becomes 0. Choice A incorrectly includes the constant 3 in the derivative, which is a mistake since constants differentiate to zero. When using the power rule, remember to eliminate all constant terms from your derivative.

To find R'(x), we need to apply the power rule to each term of the revenue function. The power rule tells us that the derivative of $x^n$ is nx^(n-1). For R(x) = $5x^6$ - $2x^3$ + 11, we differentiate term by term: $5x^6$ becomes $30x^5$, $-2x^3$ becomes $-6x^2$, and the constant 11 becomes 0. Choice A incorrectly includes the constant 11 in the derivative, which is wrong because the derivative of any constant is zero. When applying the power rule, always remember to drop constant terms from your final answer.

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 T'(t) requires applying the power rule to differentiate the temperature model. The power rule states that $\frac{d}{dx} [x^n] = n x^{n-1}$. For $T(t) = 9t^5 + t^2 - 8t + 4$, we differentiate each term: $9t^5$ becomes $45t^4$, $t^2$ becomes $2t$, $-8t$ becomes $-8$, and the constant $4$ becomes $0$. Choice D incorrectly calculates the first term as $9t^4$ instead of $45t^4$, forgetting to multiply the coefficient by the exponent. To avoid errors with the power rule, always multiply the existing coefficient by the power before reducing the exponent.