Sketching Graphs of Functions and Derivatives - AP Calculus AB
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What does $f'(x) = 0$ indicate about $f(x)$?
What does $f'(x) = 0$ indicate about $f(x)$?
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Potential extrema. Horizontal tangent lines occur at critical points.
Potential extrema. Horizontal tangent lines occur at critical points.
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What is the derivative of $f(x) = \text{arcsin}(x)$?
What is the derivative of $f(x) = \text{arcsin}(x)$?
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$f'(x) = \frac{1}{\text{√}(1-x^2)}$. Inverse trig derivative with radical denominator.
$f'(x) = \frac{1}{\text{√}(1-x^2)}$. Inverse trig derivative with radical denominator.
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State the Quotient Rule for differentiation.
State the Quotient Rule for differentiation.
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$\frac{d}{dx}[\frac{u}{v}] = \frac{u'v - uv'}{v^2}$. Low d-high minus high d-low over low squared.
$\frac{d}{dx}[\frac{u}{v}] = \frac{u'v - uv'}{v^2}$. Low d-high minus high d-low over low squared.
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What is the derivative of $f(x) = \text{cot}(x)$?
What is the derivative of $f(x) = \text{cot}(x)$?
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$f'(x) = -\text{csc}^2(x)$. Derivative of cotangent is negative cosecant squared.
$f'(x) = -\text{csc}^2(x)$. Derivative of cotangent is negative cosecant squared.
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Differentiate $f(x) = \frac{1}{x}$.
Differentiate $f(x) = \frac{1}{x}$.
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$f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.
$f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.
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Identify the concavity of $f(x) = x^4$.
Identify the concavity of $f(x) = x^4$.
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Concave up for all $x$. $f''(x) = 12x^2 \geq 0$ for all real x.
Concave up for all $x$. $f''(x) = 12x^2 \geq 0$ for all real x.
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Identify the critical points of $f(x) = x^3 - 3x$.
Identify the critical points of $f(x) = x^3 - 3x$.
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$x = 0, x = \text{±}\frac{\text{√}3}{3}$. Set $f'(x) = 3x^2 - 3 = 0$ and solve for x.
$x = 0, x = \text{±}\frac{\text{√}3}{3}$. Set $f'(x) = 3x^2 - 3 = 0$ and solve for x.
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Find $f'(x)$ for $f(x) = \text{ln}(x)$.
Find $f'(x)$ for $f(x) = \text{ln}(x)$.
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$f'(x) = \frac{1}{x}$. The natural logarithm's derivative is $\frac{1}{x}$.
$f'(x) = \frac{1}{x}$. The natural logarithm's derivative is $\frac{1}{x}$.
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What is the inverse function derivative formula?
What is the inverse function derivative formula?
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$[f^{-1}]'(x) = \frac{1}{f'(f^{-1}(x))}$. Reciprocal of derivative at corresponding point.
$[f^{-1}]'(x) = \frac{1}{f'(f^{-1}(x))}$. Reciprocal of derivative at corresponding point.
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What is $f'(x)$ for $f(x) = \text{sin}(x)$?
What is $f'(x)$ for $f(x) = \text{sin}(x)$?
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$f'(x) = \text{cos}(x)$. Derivative of sine is cosine.
$f'(x) = \text{cos}(x)$. Derivative of sine is cosine.
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Which test identifies concavity?
Which test identifies concavity?
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Second Derivative Test. Examines sign of $f''(x)$ to determine concavity.
Second Derivative Test. Examines sign of $f''(x)$ to determine concavity.
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What is $f'(x)$ for $f(x) = x^5$?
What is $f'(x)$ for $f(x) = x^5$?
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$f'(x) = 5x^4$. Power rule: bring down 5, subtract 1 from exponent.
$f'(x) = 5x^4$. Power rule: bring down 5, subtract 1 from exponent.
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State the Chain Rule for differentiation.
State the Chain Rule for differentiation.
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$\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$. Differentiate outer function times inner derivative.
$\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$. Differentiate outer function times inner derivative.
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State the Power Rule for differentiation.
State the Power Rule for differentiation.
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$\frac{d}{dx}x^n = nx^{n-1}$. Multiply by exponent, reduce exponent by 1.
$\frac{d}{dx}x^n = nx^{n-1}$. Multiply by exponent, reduce exponent by 1.
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What is the second derivative of $f(x) = 3x^4$?
What is the second derivative of $f(x) = 3x^4$?
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$f''(x) = 36x^2$. Apply power rule twice: $f'(x) = 12x^3$, then again.
$f''(x) = 36x^2$. Apply power rule twice: $f'(x) = 12x^3$, then again.
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Calculate $f'(x)$ for $f(x) = \text{cos}(x)$.
Calculate $f'(x)$ for $f(x) = \text{cos}(x)$.
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$f'(x) = -\text{sin}(x)$. Derivative of cosine is negative sine.
$f'(x) = -\text{sin}(x)$. Derivative of cosine is negative sine.
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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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$f'(x) = e^x$. The exponential function is its own derivative.
$f'(x) = e^x$. The exponential function is its own derivative.
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What is the Product Rule for differentiation?
What is the Product Rule for differentiation?
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$\frac{d}{dx}[uv] = u'v + uv'$. Sum of each function times the other's derivative.
$\frac{d}{dx}[uv] = u'v + uv'$. Sum of each function times the other's derivative.
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Differentiate $f(x) = 7x^3 - 2x$.
Differentiate $f(x) = 7x^3 - 2x$.
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$f'(x) = 21x^2 - 2$. Apply power rule to each term separately.
$f'(x) = 21x^2 - 2$. Apply power rule to each term separately.
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What is the derivative of a constant $c$?
What is the derivative of a constant $c$?
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$0$. Constants have zero rate of change.
$0$. Constants have zero rate of change.
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Find the critical points of $f(x) = x^2 - 6x + 8$.
Find the critical points of $f(x) = x^2 - 6x + 8$.
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$x = 3$. Set $f'(x) = 2x - 6 = 0$ and solve.
$x = 3$. Set $f'(x) = 2x - 6 = 0$ and solve.
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Find $f'(x)$ for $f(x) = \text{arccot}(x)$.
Find $f'(x)$ for $f(x) = \text{arccot}(x)$.
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$f'(x) = -\frac{1}{1+x^2}$. Negative of arctangent derivative.
$f'(x) = -\frac{1}{1+x^2}$. Negative of arctangent derivative.
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What is $f'(x)$ for $f(x) = \text{arctan}(x)$?
What is $f'(x)$ for $f(x) = \text{arctan}(x)$?
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$f'(x) = \frac{1}{1+x^2}$. Inverse tangent derivative with squared denominator.
$f'(x) = \frac{1}{1+x^2}$. Inverse tangent derivative with squared denominator.
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Calculate $f'(x)$ for $f(x) = \text{arccos}(x)$.
Calculate $f'(x)$ for $f(x) = \text{arccos}(x)$.
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$f'(x) = -\frac{1}{\text{√}(1-x^2)}$. Negative of arcsine derivative.
$f'(x) = -\frac{1}{\text{√}(1-x^2)}$. Negative of arcsine derivative.
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Find $f'(x)$ for $f(x) = \text{csc}(x)$.
Find $f'(x)$ for $f(x) = \text{csc}(x)$.
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$f'(x) = -\text{csc}(x)\text{cot}(x)$. Derivative involves negative cotangent cosecant.
$f'(x) = -\text{csc}(x)\text{cot}(x)$. Derivative involves negative cotangent cosecant.
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Calculate $f'(x)$ for $f(x) = \frac{1}{2}x^4$.
Calculate $f'(x)$ for $f(x) = \frac{1}{2}x^4$.
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$f'(x) = 2x^3$. Apply power rule: $4 \cdot \frac{1}{2} \cdot x^3 = 2x^3$.
$f'(x) = 2x^3$. Apply power rule: $4 \cdot \frac{1}{2} \cdot x^3 = 2x^3$.
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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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$f'(x) = 2x$. Power rule: bring down exponent, subtract 1.
$f'(x) = 2x$. Power rule: bring down exponent, subtract 1.
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What is the derivative of $f(x) = \text{tan}(x)$?
What is the derivative of $f(x) = \text{tan}(x)$?
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$f'(x) = \text{sec}^2(x)$. Derivative of tangent is secant squared.
$f'(x) = \text{sec}^2(x)$. Derivative of tangent is secant squared.
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Which test uses $f''(x)$ to find extrema?
Which test uses $f''(x)$ to find extrema?
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Second Derivative Test. Uses second derivative to classify critical points.
Second Derivative Test. Uses second derivative to classify critical points.
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Find $f'(x)$ for $f(x) = \text{ln}(x)$.
Find $f'(x)$ for $f(x) = \text{ln}(x)$.
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$f'(x) = \frac{1}{x}$. The natural logarithm's derivative is $\frac{1}{x}$.
$f'(x) = \frac{1}{x}$. The natural logarithm's derivative is $\frac{1}{x}$.
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