Applying the Power Rule - AP Calculus BC
Card 1 of 30
What is the derivative of $x^1$?
What is the derivative of $x^1$?
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$1$. The derivative of $x$ (first power) is 1.
$1$. The derivative of $x$ (first power) is 1.
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What is the derivative of $x^{10}$?
What is the derivative of $x^{10}$?
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$10x^9$. Power rule: $10 \cdot x^{10-1} = 10x^9$.
$10x^9$. Power rule: $10 \cdot x^{10-1} = 10x^9$.
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Differentiate: $f(x) = 4x^{2.3}$.
Differentiate: $f(x) = 4x^{2.3}$.
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$f'(x) = 9.2x^{1.3}$. Factor out 4, apply power rule: $4 \cdot 2.3x^{1.3}$.
$f'(x) = 9.2x^{1.3}$. Factor out 4, apply power rule: $4 \cdot 2.3x^{1.3}$.
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What is the derivative of $x^{\frac{5}{3}}$?
What is the derivative of $x^{\frac{5}{3}}$?
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$\frac{5}{3}x^{\frac{2}{3}}$. Power rule: $\frac{5}{3} \cdot x^{\frac{5}{3}-1} = \frac{5}{3}x^{\frac{2}{3}}$.
$\frac{5}{3}x^{\frac{2}{3}}$. Power rule: $\frac{5}{3} \cdot x^{\frac{5}{3}-1} = \frac{5}{3}x^{\frac{2}{3}}$.
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Find the derivative of $x^{\frac{1}{2}}$.
Find the derivative of $x^{\frac{1}{2}}$.
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$\frac{1}{2}x^{-\frac{1}{2}}$. Power rule: $\frac{1}{2} \cdot x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}}$.
$\frac{1}{2}x^{-\frac{1}{2}}$. Power rule: $\frac{1}{2} \cdot x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}}$.
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Differentiate: $f(x) = 3x^{\frac{2}{3}}$.
Differentiate: $f(x) = 3x^{\frac{2}{3}}$.
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$f'(x) = 2x^{-\frac{1}{3}}$. Factor out 3, apply power rule: $3 \cdot \frac{2}{3}x^{\frac{2}{3}-1}$.
$f'(x) = 2x^{-\frac{1}{3}}$. Factor out 3, apply power rule: $3 \cdot \frac{2}{3}x^{\frac{2}{3}-1}$.
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What is the derivative of $x^5$?
What is the derivative of $x^5$?
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$5x^4$. Power rule: $n \cdot x^{n-1} = 5 \cdot x^{5-1} = 5x^4$.
$5x^4$. Power rule: $n \cdot x^{n-1} = 5 \cdot x^{5-1} = 5x^4$.
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Differentiate: $f(x) = 6x^2$.
Differentiate: $f(x) = 6x^2$.
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$f'(x) = 12x$. Factor out 6, apply power rule: $6 \cdot 2x^1$.
$f'(x) = 12x$. Factor out 6, apply power rule: $6 \cdot 2x^1$.
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Differentiate: $f(x) = 7x^4$.
Differentiate: $f(x) = 7x^4$.
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$f'(x) = 28x^3$. Factor out constant 7, then apply power rule: $7 \cdot 4x^3$.
$f'(x) = 28x^3$. Factor out constant 7, then apply power rule: $7 \cdot 4x^3$.
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Differentiate: $f(x) = x^3$.
Differentiate: $f(x) = x^3$.
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$f'(x) = 3x^2$. Apply power rule: bring down exponent 3, subtract 1 from exponent.
$f'(x) = 3x^2$. Apply power rule: bring down exponent 3, subtract 1 from exponent.
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Differentiate: $f(x) = -4x^6$.
Differentiate: $f(x) = -4x^6$.
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$f'(x) = -24x^5$. Factor out $-4$, apply power rule: $-4 \cdot 6x^5$.
$f'(x) = -24x^5$. Factor out $-4$, apply power rule: $-4 \cdot 6x^5$.
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What is the derivative of a constant $c$?
What is the derivative of a constant $c$?
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$0$. The derivative of any constant is always zero.
$0$. The derivative of any constant is always zero.
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Differentiate: $f(x) = x^{\frac{4}{5}}$.
Differentiate: $f(x) = x^{\frac{4}{5}}$.
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$f'(x) = \frac{4}{5}x^{-\frac{1}{5}}$. Power rule: $\frac{4}{5} \cdot x^{\frac{4}{5}-1} = \frac{4}{5}x^{-\frac{1}{5}}$.
$f'(x) = \frac{4}{5}x^{-\frac{1}{5}}$. Power rule: $\frac{4}{5} \cdot x^{\frac{4}{5}-1} = \frac{4}{5}x^{-\frac{1}{5}}$.
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What is the derivative of $x^{2.7}$?
What is the derivative of $x^{2.7}$?
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$2.7x^{1.7}$. Power rule: $2.7 \cdot x^{2.7-1} = 2.7x^{1.7}$.
$2.7x^{1.7}$. Power rule: $2.7 \cdot x^{2.7-1} = 2.7x^{1.7}$.
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Differentiate: $f(x) = 7x^{0}$.
Differentiate: $f(x) = 7x^{0}$.
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$f'(x) = 0$. Since $7x^0 = 7$ (a constant), its derivative is 0.
$f'(x) = 0$. Since $7x^0 = 7$ (a constant), its derivative is 0.
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What is the derivative of $x^{\frac{1}{3}}$?
What is the derivative of $x^{\frac{1}{3}}$?
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$\frac{1}{3}x^{-\frac{2}{3}}$. Power rule: $\frac{1}{3} \cdot x^{\frac{1}{3}-1} = \frac{1}{3}x^{-\frac{2}{3}}$.
$\frac{1}{3}x^{-\frac{2}{3}}$. Power rule: $\frac{1}{3} \cdot x^{\frac{1}{3}-1} = \frac{1}{3}x^{-\frac{2}{3}}$.
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Differentiate: $f(x) = x^{-\frac{1}{2}}$.
Differentiate: $f(x) = x^{-\frac{1}{2}}$.
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$f'(x) = -\frac{1}{2}x^{-\frac{3}{2}}$. Power rule: $-\frac{1}{2} \cdot x^{-\frac{1}{2}-1} = -\frac{1}{2}x^{-\frac{3}{2}}$.
$f'(x) = -\frac{1}{2}x^{-\frac{3}{2}}$. Power rule: $-\frac{1}{2} \cdot x^{-\frac{1}{2}-1} = -\frac{1}{2}x^{-\frac{3}{2}}$.
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Differentiate: $f(x) = 8x^{3}$.
Differentiate: $f(x) = 8x^{3}$.
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$f'(x) = 24x^{2}$. Factor out 8, apply power rule: $8 \cdot 3x^2$.
$f'(x) = 24x^{2}$. Factor out 8, apply power rule: $8 \cdot 3x^2$.
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Differentiate: $f(x) = x^{-10}$.
Differentiate: $f(x) = x^{-10}$.
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$f'(x) = -10x^{-11}$. Power rule: $-10 \cdot x^{-10-1} = -10x^{-11}$.
$f'(x) = -10x^{-11}$. Power rule: $-10 \cdot x^{-10-1} = -10x^{-11}$.
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Differentiate: $f(x) = -3x^{4.5}$.
Differentiate: $f(x) = -3x^{4.5}$.
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$f'(x) = -13.5x^{3.5}$. Factor out $-3$, apply power rule: $-3 \cdot 4.5x^{3.5}$.
$f'(x) = -13.5x^{3.5}$. Factor out $-3$, apply power rule: $-3 \cdot 4.5x^{3.5}$.
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What is the derivative of $x^{0.1}$?
What is the derivative of $x^{0.1}$?
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$0.1x^{-0.9}$. Power rule: $0.1 \cdot x^{0.1-1} = 0.1x^{-0.9}$.
$0.1x^{-0.9}$. Power rule: $0.1 \cdot x^{0.1-1} = 0.1x^{-0.9}$.
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What is the derivative of $x^9$?
What is the derivative of $x^9$?
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$9x^8$. Power rule: $9 \cdot x^{9-1} = 9x^8$.
$9x^8$. Power rule: $9 \cdot x^{9-1} = 9x^8$.
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Differentiate: $f(x) = -x^8$.
Differentiate: $f(x) = -x^8$.
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$f'(x) = -8x^7$. Factor out $-1$, apply power rule: $-1 \cdot 8x^7$.
$f'(x) = -8x^7$. Factor out $-1$, apply power rule: $-1 \cdot 8x^7$.
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Differentiate: $f(x) = 10x^{0.5}$.
Differentiate: $f(x) = 10x^{0.5}$.
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$f'(x) = 5x^{-0.5}$. Factor out 10, apply power rule: $10 \cdot 0.5x^{-0.5}$.
$f'(x) = 5x^{-0.5}$. Factor out 10, apply power rule: $10 \cdot 0.5x^{-0.5}$.
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Differentiate: $f(x) = 5x^{3.5}$.
Differentiate: $f(x) = 5x^{3.5}$.
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$f'(x) = 17.5x^{2.5}$. Factor out 5, apply power rule: $5 \cdot 3.5x^{2.5}$.
$f'(x) = 17.5x^{2.5}$. Factor out 5, apply power rule: $5 \cdot 3.5x^{2.5}$.
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State the derivative of $x^{\frac{3}{2}}$.
State the derivative of $x^{\frac{3}{2}}$.
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$\frac{3}{2}x^{\frac{1}{2}}$. Power rule: $\frac{3}{2} \cdot x^{\frac{3}{2}-1} = \frac{3}{2}x^{\frac{1}{2}}$.
$\frac{3}{2}x^{\frac{1}{2}}$. Power rule: $\frac{3}{2} \cdot x^{\frac{3}{2}-1} = \frac{3}{2}x^{\frac{1}{2}}$.
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What is the derivative of $x^{-5}$?
What is the derivative of $x^{-5}$?
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$-5x^{-6}$. Power rule: $-5 \cdot x^{-5-1} = -5x^{-6}$.
$-5x^{-6}$. Power rule: $-5 \cdot x^{-5-1} = -5x^{-6}$.
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Differentiate: $f(x) = -5x^{6}$.
Differentiate: $f(x) = -5x^{6}$.
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$f'(x) = -30x^{5}$. Factor out $-5$, apply power rule: $-5 \cdot 6x^5$.
$f'(x) = -30x^{5}$. Factor out $-5$, apply power rule: $-5 \cdot 6x^5$.
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What is the derivative of $x^{1.5}$?
What is the derivative of $x^{1.5}$?
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$1.5x^{0.5}$. Power rule: $1.5 \cdot x^{1.5-1} = 1.5x^{0.5}$.
$1.5x^{0.5}$. Power rule: $1.5 \cdot x^{1.5-1} = 1.5x^{0.5}$.
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What is the derivative of $x^{\frac{4}{3}}$?
What is the derivative of $x^{\frac{4}{3}}$?
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$\frac{4}{3}x^{\frac{1}{3}}$. Power rule: $\frac{4}{3} \cdot x^{\frac{4}{3}-1} = \frac{4}{3}x^{\frac{1}{3}}$.
$\frac{4}{3}x^{\frac{1}{3}}$. Power rule: $\frac{4}{3} \cdot x^{\frac{4}{3}-1} = \frac{4}{3}x^{\frac{1}{3}}$.
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