Applying the Power Rule - AP Calculus AB

1. The height is modeled by $h(x)=2x^6-5x^4+3x$. Find $h'(x)$.
   Answer: $12x^5-20x^3+3$ This problem requires applying the power rule to find the derivative of a height model expressed as a polynomial. The power rule states that for c $x^n$, the derivative is c n $x^{n-1}$, treating each term separately. Differentiating $2x^6$ yields $12x^5$, -$5x^4$ becomes -$20x^3$, and +3x turns into +3. Combining these gives h'(x) = $12x^5$ - $20x^3$ + 3. A tempting distractor is choice D, which mistakenly changes the constant term's derivative to 3x instead of 3. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.
2. A function is $q(x)=8x^2+5x^7$. Find $q'(x)$.
   Answer: $16x+35x^6$ 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.
3. A model for distance is $d(t)=2t^8-3t^2+5t^4$. What is $d'(t)$?
   Answer: $16t^7-6t+20t^3$ This problem requires applying the power rule to a polynomial with terms out of order. The power rule states that d/dt[$t^n$] = $nt^{n-1}$. For d(t) = $2t^8$ - $3t^2$ + $5t^4$, we differentiate each term independently: the derivative of $2t^8$ is 2·$8t^7$ = $16t^7$, the derivative of -$3t^2$ is -3·$2t^1$ = -6t, and the derivative of $5t^4$ is 5·$4t^3$ = $20t^3$. Therefore, d'(t) = $16t^7$ - 6t + $20t^3$. Choice E incorrectly differentiates the last term as $5t^4$ instead of $20t^3$, keeping the original term unchanged. When differentiating polynomials, treat each term separately and don't be distracted by their order in the expression.
4. A population model is $P(t)=2t^9+6t^2-4t$. What is $P'(t)$?
   Answer: $18t^8+12t-4$ This problem involves applying the power rule to differentiate a population model. The power rule states that if f(t) = $t^n$, then f'(t) = $nt^{n-1}$. For P(t) = $2t^9$ + $6t^2$ - 4t, we differentiate each term: the derivative of $2t^9$ is 9·$2t^8$ = $18t^8$, the derivative of $6t^2$ is 2·$6t^1$ = 12t, and the derivative of -4t is -4·$1t^0$ = -4. Therefore, P'(t) = $18t^8$ + 12t - 4. Choice D incorrectly writes $12t^2$ instead of 12t, adding an extra power to the middle term. When applying the power rule, always reduce the exponent by exactly one.
5. The cost function is $C(n)=9n^3+2n^2-n+6$. What is $C'(n)$?
   Answer: $27n^2+4n-1$ 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.
6. A polynomial is $f(x)=3x^7+5x^4-2$. What is $f'(x)$?
   Answer: $21x^6+20x^3$ This problem asks us to apply the power rule to find f'(x) for the polynomial f(x) = $3x^7$ + $5x^4$ - 2. Using the power rule, we differentiate each term: $3x^7$ becomes 3·$7x^6$ = $21x^6$, $5x^4$ becomes 5·$4x^3$ = $20x^3$, and the constant -2 becomes 0. Thus, f'(x) = $21x^6$ + $20x^3$. Choice C incorrectly shows $3x^6$ + $5x^3$, which results from forgetting to multiply by the original exponents. When applying the power rule, always multiply the coefficient by the exponent before reducing the exponent by 1.
7. A polynomial is $u(t)=14t^3-5t^7+6t^2$. Find $u'(t)$.
   Answer: $42t^2-35t^6+12t$ This problem requires applying the power rule to find the derivative of the given polynomial u(t). According to the power rule, the derivative of c $t^n$ is c n $t^{n-1}$, applied term by term. For $14t^3$, it becomes $42t^2$; for -$5t^7$, it is -$35t^6$; and +$6t^2$ differentiates to +12t. Thus, u'(t) = $42t^2$ - $35t^6$ + 12t. A tempting distractor is choice D, which wrongly uses +$12t^2$ instead of +12t. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.
8. At which of the following values of $x$ does the graph of $f(x) = \frac{1}{3}x^3 - \frac{5}{2}x^2 + 6x - 1$ have a horizontal tangent line?
   Answer: $x=3$ A horizontal tangent line occurs where the derivative is equal to zero. First, find the derivative of $f(x)$ using the power rule: $f'(x) = x^2 - 5x + 6$. Set the derivative equal to zero and solve for $x$: $x^2 - 5x + 6 = 0$. Factoring the quadratic gives $(x-2)(x-3) = 0$. The solutions are $x=2$ and $x=3$. Of the choices given, $x=3$ is an answer.
9. A cost function is $C(x)=7x^2-4x^5+10$. What is $C'(x)$?
   Answer: $14x-20x^4$ 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.
10. Let $g(x) = 2x^5 - 4x^3 + 3x$. Find $g''(x)$.
    Answer: $40x^3 - 24x$ To find the second derivative, we must first find the first derivative. Using the power rule, $g'(x) = 10x^4 - 12x^2 + 3$. Now, differentiate $g'(x)$ to find $g''(x)$: $g''(x) = 10(4x^3) - 12(2x) + 0 = 40x^3 - 24x$.
11. An object's position is $s(x)=9x^4+2x^3-5$. Find $s'(x)$.
    Answer: $36x^3+6x^2$ This problem requires applying the power rule to find the derivative of a polynomial representing an object's position. The power rule dictates that for c $x^n$, the derivative is c n $x^{n-1}$, handling each term separately. Differentiating $9x^4$ gives $36x^3$, while $2x^3$ becomes $6x^2$, and the constant -5 becomes 0. Combining these yields s'(x) = $36x^3$ + $6x^2$. A tempting distractor is choice D, which wrongly includes the constant -5, but constants disappear in derivatives. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.
12. The derivative of $f(x) = \frac{4x^5 - 6x^3}{2x^2}$ for $x \neq 0$ is
    Answer: $6x^2 - 3$ 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: $f'(x) = 2(3x^2) - 3(1) = 6x^2 - 3$.
13. The revenue is modeled by $R(x)=5x^7+4x^2-9$. What is $R'(x)$?
    Answer: $35x^6+8x$ This problem involves applying the power rule to find the derivative of a revenue function. The power rule states that if f(x) = $x^n$, then f'(x) = $nx^{n-1}$. For R(x) = $5x^7$ + $4x^2$ - 9, we differentiate each term separately: the derivative of $5x^7$ is 7·$5x^6$ = $35x^6$, the derivative of $4x^2$ is 2·$4x^1$ = 8x, and the derivative of the constant -9 is 0. Thus, R'(x) = $35x^6$ + 8x. Choice A incorrectly includes the constant -9, failing to apply the rule that constants have zero derivatives. When differentiating polynomials, systematically apply the power rule to each term and eliminate all constant terms.
14. Let $g(x)=7x^6-x^5+11x^2$. What is $g'(x)$?
    Answer: $42x^5-5x^4+22x$ This problem requires applying the power rule to find g'(x) where g(x) = $7x^6$ - $x^5$ + $11x^2$. The power rule tells us that d/dx[$x^n$] = $nx^{n-1}$. For each term: the derivative of $7x^6$ is 6·$7x^5$ = $42x^5$, the derivative of -$x^5$ is 5·(-1)$x^4$ = -$5x^4$, and the derivative of $11x^2$ is 2·$11x^1$ = 22x. Thus, g'(x) = $42x^5$ - $5x^4$ + 22x. Choice C incorrectly gives the coefficient of $x^4$ as -1 instead of -5, forgetting to multiply by the exponent 5. Always remember to multiply the coefficient by the original exponent when differentiating.
15. A displacement function is $d(t)=t^5+3t^4-2t^2$. What is $d'(t)$?
    Answer: $5t^4+12t^3-4t$ This problem requires applying the power rule to find the derivative of a displacement function polynomial. The power rule states that for c $t^n$, the derivative is c n $t^{n-1}$, treating terms separately. Differentiating $t^5$ gives $5t^4$, +$3t^4$ becomes +$12t^3$, and -$2t^2$ turns into -4t. Combining yields d'(t) = $5t^4$ + $12t^3$ - 4t. A tempting distractor is choice D, which incorrectly uses -$4t^2$ instead of -4t. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.
16. A function is $q(t)=t^{10}-6t^3+2t^2-1$. What is $q'(t)$?
    Answer: $10t^9-18t^2+4t$ To find q'(t), we must apply the power rule to differentiate each term of the function. The power rule tells us that d/dt[$t^n$] = $nt^{n-1}$. For q(t) = $t^{10}$ - $6t^3$ + $2t^2$ - 1, we get: $t^{10}$ becomes 10·$t^9$ = $10t^9$, -$6t^3$ becomes 3·(-6)$t^2$ = -$18t^2$, $2t^2$ becomes 2·$2t^1$ = 4t, and the constant -1 becomes 0. Choice E ($10t^9$ - $6t^2$ + 4t) incorrectly differentiates -$6t^3$ as -$6t^2$, forgetting to multiply by the exponent 3. Always remember to multiply the coefficient by the original exponent before reducing the exponent by 1.
17. A polynomial is $r(x)=x^4-12x+6x^2$. What is $r'(x)$?
    Answer: $4x^3-12+12x$ This problem requires applying the power rule to find the derivative of the given polynomial r(x). The power rule indicates that for c $x^n$, the derivative is c n $x^{n-1}$, handling each term individually. Differentiating $x^4$ gives $4x^3$, -12x becomes -12, and +$6x^2$ turns into +12x. Combining yields r'(x) = $4x^3$ - 12 + 12x. A tempting distractor is choice D, which incorrectly uses +6x instead of +12x. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.
18. The position of a particle moving on the x-axis is given by $x(t) = t^3 - 6t^2 + 5$ for time $t \ge 0$. What is the velocity of the particle at time $t=4$?
    Answer: 0 Velocity is the derivative of the position function. Let $v(t)$ be the velocity. Then $v(t) = x'(t)$. Using the power rule, $x'(t) = 3t^2 - 12t$. To find the velocity at $t=4$, substitute $t=4$ into the derivative: $v(4) = 3(4)^2 - 12(4) = 3(16) - 48 = 48 - 48 = 0$.
19. The function is $f(x)=3x^8-10x^4+x^3$. What is $f'(x)$?
    Answer: $24x^7-40x^3+3x^2$ 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.
20. A function is $g(x)=12x^3-6x^2+4$. What is $g'(x)$?
    Answer: $36x^2-12x$ This problem requires applying the power rule to find the derivative of the given polynomial function g(x). The power rule states that for c $x^n$, the derivative is c n $x^{n-1}$, treating terms individually. Differentiating $12x^3$ gives $36x^2$, -$6x^2$ becomes -12x, and +4 turns to 0. Combining these yields g'(x) = $36x^2$ - 12x. A tempting distractor is choice D, which incorrectly includes the constant 4 in the derivative. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

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