First, simplify the expression for $$f(x)$$ by dividing each term in the numerator by the denominator: $$f(x) = \frac{4x^5}{2x^2} - \frac{6x^3}{2x^2} = 2x^3 - 3x$$. Now, differentiate the simplified polynomial using the power rule: $$f'(x) = 2(3x^2) - 3(1) = 6x^2 - 3$$.

The derivative of a polynomial is found by applying the power rule, $$d/dx(x^n) = nx^{n-1}$$, to each term. For $$f(x) = 5x^4 - 3x^2 + 2x^1 - 7x^0$$, the derivative is $$f'(x) = 5(4x^{4-1}) - 3(2x^{2-1}) + 2(1x^{1-1}) - 7(0x^{0-1}) = 20x^3 - 6x + 2$$.

This problem asks us to apply the power rule to find f'(x) where f(x) = $3x^8$ - $10x^4$ + $x^3$. The power rule states that the derivative of $x^n$ is nx^(n-1). Applying this to each term: the derivative of $3x^8$ is $8·3x^7$ = $24x^7$, the derivative of $-10x^4$ is $4·(-10)x^3$ = $-40x^3$, and the derivative of $x^3$ is $3x^2$. Thus, f'(x) = $24x^7$ - $40x^3$ + $3x^2$. Choice E incorrectly gives the coefficient of $x^2$ as 1 instead of 3, forgetting to multiply by the original exponent. Remember that the power rule requires multiplying by the exponent, even when the original coefficient is 1.

This problem asks us to apply the power rule to find the derivative of a cost function. The power rule tells us that the derivative of $x^n$ is nx^(n-1). For C(x) = $9x^4$ - $2x^3$ + x, 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 x (which is $x^1$) is $1·x^0$ = 1. Thus, C'(x) = $36x^3$ - $6x^2$ + 1. Choice E incorrectly omits the derivative of the x term, forgetting that x has a derivative of 1. Always include the derivatives of all terms, even when the coefficient is 1.

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.

This problem requires applying the power rule to find the derivative of a cost function that is a polynomial. Per the power rule, the derivative of c $x^n$ is c n $x^{n-1}$, applied term by term. For $7x^2$, it is 14x; for $-4x^5$, it becomes $-20x^4$; and +10 differentiates to 0. Thus, C'(x) = 14x - $20x^4$. A tempting distractor is choice D, which wrongly retains the constant 10, but constants differentiate to zero. 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 derivative of a revenue model polynomial. Per the power rule, the derivative of c $x^n$ is c n $x^{n-1}$, applied to each term. For $3x^8$, it becomes $24x^7$; for $-2x^4$, it is $-8x^3$; and +x $(x^1$) differentiates to +1. Thus, R'(x) = $24x^7$ - $8x^3$ + 1. A tempting distractor is choice D, which wrongly changes the constant 1 to x. 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 position function. The power rule states that if f(x) = $x^n$, then f'(x) = nx^(n-1). For s(t) = $4t^5$ - $3t^2$ + 7, we differentiate each term: the derivative of $4t^5$ is $5·4t^4$ = $20t^4$, the derivative of $-3t^2$ is $2·(-3)t^1$ = -6t, and the derivative of the constant 7 is 0. Therefore, s'(t) = $20t^4$ - 6t. Choice E incorrectly keeps the constant term 7, failing to recognize that constants have zero derivatives. When differentiating polynomials, apply the power rule term by term and remember that constants disappear.

This problem asks us to apply the power rule to differentiate a height function. The power rule states that the derivative of $t^n$ is nt^(n-1). For H(t) = $15t^4$ + $3t^3$ - 2, we differentiate term by term: the derivative of $15t^4$ is $4·15t^3$ = $60t^3$, the derivative of $3t^3$ is $3·3t^2$ = $9t^2$, and the derivative of the constant -2 is 0. Therefore, H'(t) = $60t^3$ + $9t^2$. Choice E incorrectly retains the constant -2, not recognizing that constants have zero derivatives. When differentiating, always remember that constant terms vanish completely.