Linearity rules allow independent differentiation of terms in F(x) = $3x^3$ - $4x^2$ + 5x - 6. Derivative of $3x^3$ is $9x^2$, of $-4x^2$ is -8x, of 5x is 5, of -6 is 0. Thus, F'(x) = $9x^2$ - 8x + 5. A common misuse is subtracting instead of adding derivatives in sums. Constants are often incorrectly included. Decompose using linearity and apply power rule to each term for accuracy.

The linearity rules facilitate term-by-term differentiation for v(t) = $6t^8$ - $2t^7$ + $5t^2$ - 9t + 1. Derivative of $6t^8$ is $48t^7$, of $-2t^7$ is $-14t^6$, of $5t^2$ is 10t, of -9t is -9, of +1 is 0. Thus, v'(t) = $48t^7$ - $14t^6$ + 10t - 9. A common misuse is miscalculating coefficients for high powers, like 6*8=48. Signs need precise tracking. For complex polynomials, apply power rule via linearity to each term individually.

Linearity enables term-by-term differentiation using sum, difference, and constant multiple rules for g(x). For g(x) = $-8x^4$ + $6x^3$ + 2x - 11, apply the power rule accordingly. The derivative of $-8x^4$ is $-32x^3$, of $6x^3$ is $18x^2$, of 2x is 2, and of -11 is 0. So, g'(x) = $-32x^3$ + $18x^2$ + 2, matching choice A. A common misuse is not reducing the power, like $x^4$ to $-32x^4$ instead of $-32x^3$. Another error is adding rather than subtracting for difference terms. A transferable strategy is to isolate each term, use d/dx (c $x^n$) = c n $x^{n-1}$, ensure constants are zero, and combine using linearity.

The volume model's derivative uses linearity to treat terms separately with differences and multiples. For V(x) = $15x^2$ - 6x + 20, apply the power rule. The derivative of $15x^2$ is 30x, of -6x is -6, and of 20 is 0. Thus, V'(x) = 30x - 6, corresponding to choice A. Common misuse includes not multiplying by 2 for quadratic terms. Another is deriving constants. A transferable strategy is to focus on quadratic forms, use d/dx (c $x^n$) = c n $x^{n-1}$, and assemble with original operations.

Using linearity, differentiate L(t) = $-4t^5$ + $9t^4$ - $3t^3$ + 2t by applying the power rule to each term. Derivative of $-4t^5$ is $-20t^4$, of $9t^4$ is $36t^3$, of $-3t^3$ is $-9t^2$, of 2t is 2. Thus, L'(t) = $-20t^4$ + $36t^3$ - $9t^2$ + 2. A common misuse is forgetting to multiply negative coefficients correctly, leading to sign errors. Overlooking the linear term's derivative is another pitfall. For polynomials, leverage linearity to handle sums and constants systematically.

Linearity enables differentiating K(x) = $-x^7$ + $2x^6$ - $3x^2$ + 4x - 12 term by term. Derivative of $-x^7$ is $-7x^6$, of $2x^6$ is $12x^5$, of $-3x^2$ is -6x, of 4x is 4, of -12 is 0. Thus, K'(x) = $-7x^6$ + $12x^5$ - 6x + 4. A common misuse is not multiplying by the exponent for unit coefficients, like $-x^7$ becoming $-x^6$. Signs and constants require attention. Use linearity to decompose and apply power rule for each power.

Linearity enables computing W'(x) for W(x) = $-9x^3$ + $4x^2$ - 2x + 16 by differentiating each term. Derivative of $-9x^3$ is $-27x^2$, of $4x^2$ is 8x, of -2x is -2, of +16 is 0. Thus, W'(x) = $-27x^2$ + 8x - 2. A common misuse is forgetting to multiply by the exponent, such as $-9x^2$ for $-9x^3$. Constant derivatives are zero, but sometimes added erroneously. Use linearity to break down and apply power rule term by term for accuracy.

Linearity permits term-by-term computation for R(x) = $9x^5$ - $6x^2$ + 2x + 1. Derivative of $9x^5$ is $45x^4$, of $-6x^2$ is -12x, of 2x is 2, of +1 is 0. Thus, R'(x) = $45x^4$ - 12x + 2. A common misuse is forgetting the constant multiple in the power rule, like $5x^4$ for $x^5$. Constants should not appear in derivatives. Select rules by identifying polynomial structure and applying linearity with power rule per term.

The sum and constant multiple rules under linearity simplify S(t) = $-14t^2$ + 7t + 2. Derivative of $-14t^2$ is -28t, of 7t is 7, of +2 is 0. Thus, S'(t) = -28t + 7. A common misuse is forgetting the negative on quadratic terms. Linear coefficients might be mishandled. For quadratics, use linearity to differentiate powers separately, ignoring constants.

Linearity enables us to differentiate each term in Q(x) = $2x^3$ - $7x^2$ + 0x + 19 independently. The derivative is $6x^2$ for $2x^3$, -14x for $-7x^2$, 0 for 0x, and 0 for +19. Thus, Q'(x) = $6x^2$ - 14x. A common rule misuse is treating the zero coefficient term as contributing something, but it vanishes. Miscalculating coefficients like 2*3=6 is another error. Always decompose polynomials using sum rules and apply power rule term by term for correct derivatives.