Connecting a Function and Its Derivatives - AP Calculus BC
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What is the second derivative of $f(x) = \ln(x)$?
What is the second derivative of $f(x) = \ln(x)$?
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$f''(x) = -\frac{1}{x^2}$. Derivative of $\frac{1}{x}$ applied to $\ln(x)$.
$f''(x) = -\frac{1}{x^2}$. Derivative of $\frac{1}{x}$ applied to $\ln(x)$.
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Determine $f'(x)$ for $f(x) = e^x$.
Determine $f'(x)$ for $f(x) = e^x$.
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$f'(x) = e^x$. The exponential function is its own derivative.
$f'(x) = e^x$. The exponential function is its own derivative.
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What does the concavity test involve?
What does the concavity test involve?
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Analyzing $f''(x)$ to determine concavity. Examining where $f''(x)$ changes sign.
Analyzing $f''(x)$ to determine concavity. Examining where $f''(x)$ changes sign.
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Determine $f''(x)$ for $f(x) = \text{cos}(2x)$.
Determine $f''(x)$ for $f(x) = \text{cos}(2x)$.
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$f''(x) = -4\text{cos}(2x)$. Apply chain rule: derivative of $\cos(2x)$ is $-2\sin(2x)$, then again.
$f''(x) = -4\text{cos}(2x)$. Apply chain rule: derivative of $\cos(2x)$ is $-2\sin(2x)$, then again.
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Find $f''(x)$ for $f(x) = x^4$.
Find $f''(x)$ for $f(x) = x^4$.
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$f''(x) = 12x^2$. Apply power rule: $f'(x) = 4x^3$, then $f''(x) = 12x^2$.
$f''(x) = 12x^2$. Apply power rule: $f'(x) = 4x^3$, then $f''(x) = 12x^2$.
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What is the second derivative of $f(x) = \text{e}^x$?
What is the second derivative of $f(x) = \text{e}^x$?
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$f''(x) = \text{e}^x$. Exponential function equals its own second derivative.
$f''(x) = \text{e}^x$. Exponential function equals its own second derivative.
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What is the second derivative of $f(x) = \text{cos}(x)$?
What is the second derivative of $f(x) = \text{cos}(x)$?
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$f''(x) = -\text{cos}(x)$. Derivative of cosine is $-\sin(x)$; derivative again gives $-\cos(x)$.
$f''(x) = -\text{cos}(x)$. Derivative of cosine is $-\sin(x)$; derivative again gives $-\cos(x)$.
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Find $f'(x)$ for $f(x) = \frac{1}{x^2}$.
Find $f'(x)$ for $f(x) = \frac{1}{x^2}$.
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$f'(x) = -\frac{2}{x^3}$. Rewrite as $x^{-2}$ and apply power rule.
$f'(x) = -\frac{2}{x^3}$. Rewrite as $x^{-2}$ and apply power rule.
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Calculate $f''(x)$ for $f(x) = \text{sin}(x)$.
Calculate $f''(x)$ for $f(x) = \text{sin}(x)$.
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$f''(x) = -\text{sin}(x)$. Derivative of sine is cosine; derivative of cosine is $-\sin(x)$.
$f''(x) = -\text{sin}(x)$. Derivative of sine is cosine; derivative of cosine is $-\sin(x)$.
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What is the second derivative of $f(x) = \text{ln}(x^2)$?
What is the second derivative of $f(x) = \text{ln}(x^2)$?
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$f''(x) = -\frac{2}{x^2}$. Use chain rule: $f'(x) = \frac{2}{x}$, then apply quotient rule.
$f''(x) = -\frac{2}{x^2}$. Use chain rule: $f'(x) = \frac{2}{x}$, then apply quotient rule.
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What is the derivative of $f(x) = \text{cos}(x)$?
What is the derivative of $f(x) = \text{cos}(x)$?
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$f'(x) = -\text{sin}(x)$. Standard derivative of cosine function.
$f'(x) = -\text{sin}(x)$. Standard derivative of cosine function.
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Calculate $f'(x)$ for $f(x) = \text{tan}(x)$.
Calculate $f'(x)$ for $f(x) = \text{tan}(x)$.
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$f'(x) = \text{sec}^2(x)$. Standard derivative of tangent function.
$f'(x) = \text{sec}^2(x)$. Standard derivative of tangent function.
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Determine the first derivative of $f(x) = \text{e}^{2x}$.
Determine the first derivative of $f(x) = \text{e}^{2x}$.
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$f'(x) = 2\text{e}^{2x}$. Apply chain rule: $\frac{d}{dx}[e^{u}] = e^{u} \cdot u'$.
$f'(x) = 2\text{e}^{2x}$. Apply chain rule: $\frac{d}{dx}[e^{u}] = e^{u} \cdot u'$.
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What is the second derivative test used for?
What is the second derivative test used for?
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To determine if a critical point is a local max or min. Uses sign of $f''(c)$ to classify critical points.
To determine if a critical point is a local max or min. Uses sign of $f''(c)$ to classify critical points.
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What is the derivative of $f(x) = \text{sin}(2x)$?
What is the derivative of $f(x) = \text{sin}(2x)$?
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$f'(x) = 2\text{cos}(2x)$. Apply chain rule to $\sin(2x)$.
$f'(x) = 2\text{cos}(2x)$. Apply chain rule to $\sin(2x)$.
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What is the first derivative of $f(x) = \text{sin}^2(x)$?
What is the first derivative of $f(x) = \text{sin}^2(x)$?
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$f'(x) = 2\text{sin}(x)\text{cos}(x)$. Apply chain rule to $\sin^2(x) = (\sin(x))^2$.
$f'(x) = 2\text{sin}(x)\text{cos}(x)$. Apply chain rule to $\sin^2(x) = (\sin(x))^2$.
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What is the derivative of $f(x) = x \text{e}^x$?
What is the derivative of $f(x) = x \text{e}^x$?
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$f'(x) = \text{e}^x + x \text{e}^x$. Apply product rule: $(uv)' = u'v + uv'$.
$f'(x) = \text{e}^x + x \text{e}^x$. Apply product rule: $(uv)' = u'v + uv'$.
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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: $f'(x) = 3x^2$, then $f''(x) = 6x$.
$f''(x) = 6x$. Apply power rule twice: $f'(x) = 3x^2$, then $f''(x) = 6x$.
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Find the critical points of $f(x) = x^3 - 3x^2$.
Find the critical points of $f(x) = x^3 - 3x^2$.
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At $x = 0$ and $x = 2$. Set $f'(x) = 3x^2 - 6x = 3x(x-2) = 0$.
At $x = 0$ and $x = 2$. Set $f'(x) = 3x^2 - 6x = 3x(x-2) = 0$.
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What can be concluded if $f''(x) < 0$ at a point?
What can be concluded if $f''(x) < 0$ at a point?
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$f(x)$ is concave down. Negative second derivative means curve bends downward.
$f(x)$ is concave down. Negative second derivative means curve bends downward.
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If $f''(x) > 0$, what can be said about $f(x)$?
If $f''(x) > 0$, what can be said about $f(x)$?
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$f(x)$ is concave up. Positive second derivative means curve bends upward.
$f(x)$ is concave up. Positive second derivative means curve bends upward.
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Find $f'(x)$ for $f(x) = x^2 + 2x + 1$.
Find $f'(x)$ for $f(x) = x^2 + 2x + 1$.
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$f'(x) = 2x + 2$. Apply power rule to polynomial.
$f'(x) = 2x + 2$. Apply power rule to polynomial.
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What is the derivative of $f(x) = x^2$?
What is the derivative of $f(x) = x^2$?
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$f'(x) = 2x$. Apply power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$.
$f'(x) = 2x$. Apply power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$.
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Identify the critical points of $f(x) = x^2 - 4x + 4$.
Identify the critical points of $f(x) = x^2 - 4x + 4$.
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At $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find critical points.
At $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find critical points.
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Identify the derivative of $f(x) = \frac{1}{x^3}$.
Identify 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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What is the first derivative of $f(x) = \text{ln}(x)$?
What is the first derivative of $f(x) = \text{ln}(x)$?
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$f'(x) = \frac{1}{x}$. Standard derivative of natural logarithm function.
$f'(x) = \frac{1}{x}$. Standard derivative of natural logarithm function.
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Find $f''(x)$ for $f(x) = \frac{1}{x}$.
Find $f''(x)$ for $f(x) = \frac{1}{x}$.
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$f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, apply power rule twice.
$f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, apply power rule twice.
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State the power rule for differentiation.
State the power rule for differentiation.
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$\frac{d}{dx}[x^n] = nx^{n-1}$. Fundamental rule for differentiating polynomial terms.
$\frac{d}{dx}[x^n] = nx^{n-1}$. Fundamental rule for differentiating polynomial terms.
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What does the second derivative of a function represent?
What does the second derivative of a function represent?
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The concavity of the function. Second derivative determines curve shape and bending.
The concavity of the function. Second derivative determines curve shape and bending.
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Find the derivative of $f(x) = \text{e}^{-x}$.
Find the derivative of $f(x) = \text{e}^{-x}$.
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$f'(x) = -\text{e}^{-x}$. Apply chain rule with $u = -x$.
$f'(x) = -\text{e}^{-x}$. Apply chain rule with $u = -x$.
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