Connecting a Function and Its Derivatives - AP Calculus AB
Card 1 of 30
Find $f''(x)$ for $f(x) = x^3 - 5x$.
Find $f''(x)$ for $f(x) = x^3 - 5x$.
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$f''(x) = 6x$. First find $f'(x) = 3x^2 - 5$, then differentiate.
$f''(x) = 6x$. First find $f'(x) = 3x^2 - 5$, then differentiate.
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What is the derivative of a constant function $f(x) = c$?
What is the derivative of a constant function $f(x) = c$?
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The derivative is $0$. Constants have zero rate of change.
The derivative is $0$. Constants have zero rate of change.
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What indicates a local maximum in terms of derivatives?
What indicates a local maximum in terms of derivatives?
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$f'(x) = 0$ and $f''(x) < 0$. Critical point test: zero slope, negative concavity.
$f'(x) = 0$ and $f''(x) < 0$. Critical point test: zero slope, negative concavity.
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What is $f''(x)$ for $f(x) = \text{e}^x + x^2$?
What is $f''(x)$ for $f(x) = \text{e}^x + x^2$?
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$f''(x) = \text{e}^x + 2$. Sum rule: derivatives of $e^x$ and $x^2$ separately.
$f''(x) = \text{e}^x + 2$. Sum rule: derivatives of $e^x$ and $x^2$ separately.
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What is the first derivative of $f(x) = \frac{1}{x^2}$?
What is the first derivative of $f(x) = \frac{1}{x^2}$?
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$f'(x) = -\frac{2}{x^3}$. Rewrite as $x^{-2}$, then apply power rule.
$f'(x) = -\frac{2}{x^3}$. Rewrite as $x^{-2}$, then apply power rule.
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State the relationship between concavity and the second derivative.
State the relationship between concavity and the second derivative.
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Concave up if $f''(x) > 0$, concave down if $f''(x) < 0$. Second derivative test for concavity direction.
Concave up if $f''(x) > 0$, concave down if $f''(x) < 0$. Second derivative test for concavity direction.
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Find the second derivative: $f(x) = 2x^4 - 3x^3 + x$.
Find the second derivative: $f(x) = 2x^4 - 3x^3 + x$.
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$f''(x) = 24x^2 - 18x$. First find $f'(x) = 8x^3 - 9x^2 + 1$, then differentiate.
$f''(x) = 24x^2 - 18x$. First find $f'(x) = 8x^3 - 9x^2 + 1$, then differentiate.
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What is the derivative of $f(x) = e^x$?
What is the derivative of $f(x) = e^x$?
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The derivative is $f'(x) = e^x$. The exponential function is its own derivative.
The derivative is $f'(x) = e^x$. The exponential function is its own derivative.
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What is $f''(x)$ if $f(x) = x^5 - 3x^2 + 1$?
What is $f''(x)$ if $f(x) = x^5 - 3x^2 + 1$?
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$f''(x) = 20x^3 - 6$. First find $f'(x) = 5x^4 - 6x$, then differentiate.
$f''(x) = 20x^3 - 6$. First find $f'(x) = 5x^4 - 6x$, then differentiate.
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What is the second derivative of $f(x) = \text{sin}(x)$?
What is the second derivative of $f(x) = \text{sin}(x)$?
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$f''(x) = -\text{sin}(x)$. First derivative is $\cos(x)$, then $-\sin(x)$.
$f''(x) = -\text{sin}(x)$. First derivative is $\cos(x)$, then $-\sin(x)$.
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Determine $f'(x)$ for $f(x) = \text{cos}(x)$.
Determine $f'(x)$ for $f(x) = \text{cos}(x)$.
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$f'(x) = -\text{sin}(x)$. Standard trigonometric derivative formula.
$f'(x) = -\text{sin}(x)$. Standard trigonometric derivative formula.
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If $f''(x) = 0$, what can be inferred about $f(x)$?
If $f''(x) = 0$, what can be inferred about $f(x)$?
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$f(x)$ may have a point of inflection. Second derivative test for inflection points.
$f(x)$ may have a point of inflection. Second derivative test for inflection points.
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Evaluate the second derivative: $f(x) = 6x - 4x^2$.
Evaluate the second derivative: $f(x) = 6x - 4x^2$.
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$f''(x) = -8$. First find $f'(x) = 6 - 8x$, then differentiate.
$f''(x) = -8$. First find $f'(x) = 6 - 8x$, then differentiate.
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Determine $f''(x)$ for $f(x) = 3x^4 + x^2$.
Determine $f''(x)$ for $f(x) = 3x^4 + x^2$.
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$f''(x) = 36x^2 + 2$. First find $f'(x) = 12x^3 + 2x$, then differentiate.
$f''(x) = 36x^2 + 2$. First find $f'(x) = 12x^3 + 2x$, then differentiate.
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If $f'(x) > 0$ for all $x$, what can you say about $f(x)$?
If $f'(x) > 0$ for all $x$, what can you say about $f(x)$?
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$f(x)$ is increasing. Positive derivative indicates upward slope.
$f(x)$ is increasing. Positive derivative indicates upward slope.
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State the Power Rule for differentiation.
State the Power Rule for differentiation.
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If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Bring down the exponent and reduce power by 1.
If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Bring down the exponent and reduce power by 1.
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Determine the second derivative: $f(x) = 4x^2 + 3x$.
Determine the second derivative: $f(x) = 4x^2 + 3x$.
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$f''(x) = 8$. First derivative is $8x + 3$, then derivative is $8$.
$f''(x) = 8$. First derivative is $8x + 3$, then derivative is $8$.
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What is the second derivative of $f(x) = x^3$?
What is the second derivative of $f(x) = x^3$?
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$f''(x) = 6x$. Apply power rule twice: $3x^2$ then $6x$.
$f''(x) = 6x$. Apply power rule twice: $3x^2$ then $6x$.
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If $f'(x) < 0$ for all $x$, what is true about $f(x)$?
If $f'(x) < 0$ for all $x$, what is true about $f(x)$?
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$f(x)$ is decreasing. Negative derivative indicates downward slope.
$f(x)$ is decreasing. Negative derivative indicates downward slope.
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Evaluate $f'(x)$ for $f(x) = \text{tan}(x)$.
Evaluate $f'(x)$ for $f(x) = \text{tan}(x)$.
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$f'(x) = \text{sec}^2(x)$. Standard trigonometric derivative formula.
$f'(x) = \text{sec}^2(x)$. Standard trigonometric derivative formula.
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Determine the derivative of $f(x) = \frac{1}{x^3}$.
Determine the derivative of $f(x) = \frac{1}{x^3}$.
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$f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.
$f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.
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Evaluate $f'(x)$ for $f(x) = 5x^3 - 4x^2 + 2x - 7$.
Evaluate $f'(x)$ for $f(x) = 5x^3 - 4x^2 + 2x - 7$.
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$f'(x) = 15x^2 - 8x + 2$. Apply power rule to each term separately.
$f'(x) = 15x^2 - 8x + 2$. Apply power rule to each term separately.
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Evaluate $f'(x)$ for $f(x) = 5x^3 - 4x^2 + 2x - 7$.
Evaluate $f'(x)$ for $f(x) = 5x^3 - 4x^2 + 2x - 7$.
Tap to reveal answer
$f'(x) = 15x^2 - 8x + 2$. Apply power rule to each term separately.
$f'(x) = 15x^2 - 8x + 2$. Apply power rule to each term separately.
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Determine the derivative of $f(x) = \frac{1}{x^3}$.
Determine the derivative of $f(x) = \frac{1}{x^3}$.
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$f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.
$f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.
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Determine $f''(x)$ for $f(x) = 3x^4 + x^2$.
Determine $f''(x)$ for $f(x) = 3x^4 + x^2$.
Tap to reveal answer
$f''(x) = 36x^2 + 2$. First find $f'(x) = 12x^3 + 2x$, then differentiate.
$f''(x) = 36x^2 + 2$. First find $f'(x) = 12x^3 + 2x$, then differentiate.
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Evaluate the second derivative: $f(x) = 6x - 4x^2$.
Evaluate the second derivative: $f(x) = 6x - 4x^2$.
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$f''(x) = -8$. First find $f'(x) = 6 - 8x$, then differentiate.
$f''(x) = -8$. First find $f'(x) = 6 - 8x$, then differentiate.
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Evaluate $f'(x)$ for $f(x) = \text{tan}(x)$.
Evaluate $f'(x)$ for $f(x) = \text{tan}(x)$.
Tap to reveal answer
$f'(x) = \text{sec}^2(x)$. Standard trigonometric derivative formula.
$f'(x) = \text{sec}^2(x)$. Standard trigonometric derivative formula.
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If $f'(x) < 0$ for all $x$, what is true about $f(x)$?
If $f'(x) < 0$ for all $x$, what is true about $f(x)$?
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$f(x)$ is decreasing. Negative derivative indicates downward slope.
$f(x)$ is decreasing. Negative derivative indicates downward slope.
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What is the second derivative of $f(x) = x^3$?
What is the second derivative of $f(x) = x^3$?
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$f''(x) = 6x$. Apply power rule twice: $3x^2$ then $6x$.
$f''(x) = 6x$. Apply power rule twice: $3x^2$ then $6x$.
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State the Power Rule for differentiation.
State the Power Rule for differentiation.
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If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Bring down the exponent and reduce power by 1.
If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Bring down the exponent and reduce power by 1.
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