Introduction to Related Rates - AP Calculus AB
Card 1 of 30
Find $\frac{dV}{dt}$ if $V = \frac{4}{3}\times \pi r^3$ and $\frac{dr}{dt} = 2$ at $r = 3$.
Find $\frac{dV}{dt}$ if $V = \frac{4}{3}\times \pi r^3$ and $\frac{dr}{dt} = 2$ at $r = 3$.
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$\frac{dV}{dt} = 72\pi$. Volume formula for sphere, differentiated.
$\frac{dV}{dt} = 72\pi$. Volume formula for sphere, differentiated.
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State the formula for the derivative of a product of two functions.
State the formula for the derivative of a product of two functions.
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$(f \times g)' = f' \times g + f \times g'$. Product rule for derivatives.
$(f \times g)' = f' \times g + f \times g'$. Product rule for derivatives.
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What is the derivative of $\frac{d}{dx}(\text{sec}(x))$?
What is the derivative of $\frac{d}{dx}(\text{sec}(x))$?
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$\frac{d}{dx}(\text{sec}(x)) = \text{sec}(x)\text{tan}(x)$. Secant derivative involves secant and tangent.
$\frac{d}{dx}(\text{sec}(x)) = \text{sec}(x)\text{tan}(x)$. Secant derivative involves secant and tangent.
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Find $\frac{dV}{dt}$ for $V = \frac{4}{3}\text{π}r^3$ if $r = 6$ and $\frac{dr}{dt} = 0.5$.
Find $\frac{dV}{dt}$ for $V = \frac{4}{3}\text{π}r^3$ if $r = 6$ and $\frac{dr}{dt} = 0.5$.
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$\frac{dV}{dt} = 72\text{π}$. Sphere volume: $\frac{dV}{dt} = 4\pi r^2 \cdot 0.5 = 72\pi$.
$\frac{dV}{dt} = 72\text{π}$. Sphere volume: $\frac{dV}{dt} = 4\pi r^2 \cdot 0.5 = 72\pi$.
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What is the derivative of $\frac{d}{dx}(a^x)$?
What is the derivative of $\frac{d}{dx}(a^x)$?
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$\frac{d}{dx}(a^x) = a^x\text{ln}(a)$. General exponential function derivative.
$\frac{d}{dx}(a^x) = a^x\text{ln}(a)$. General exponential function derivative.
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Determine $\frac{dy}{dt}$ for $y = x^3$ if $x = 2$ and $\frac{dx}{dt} = 3$.
Determine $\frac{dy}{dt}$ for $y = x^3$ if $x = 2$ and $\frac{dx}{dt} = 3$.
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$\frac{dy}{dt} = 36$. Power rule: $\frac{dy}{dt} = 3x^2 \cdot 3 = 36$.
$\frac{dy}{dt} = 36$. Power rule: $\frac{dy}{dt} = 3x^2 \cdot 3 = 36$.
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What is the formula for the derivative of $\frac{d}{dx}(\cos(x))$?
What is the formula for the derivative of $\frac{d}{dx}(\cos(x))$?
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$\frac{d}{dx}(\cos(x)) = -\sin(x)$. Cosine derivative is negative sine.
$\frac{d}{dx}(\cos(x)) = -\sin(x)$. Cosine derivative is negative sine.
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Which rule is primarily used in related rates problems?
Which rule is primarily used in related rates problems?
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Chain rule. Links rates through composite functions.
Chain rule. Links rates through composite functions.
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Determine $\frac{dC}{dt}$ if $C = 2\text{π}r$ and $\frac{dr}{dt} = 5$ at $r = 4$.
Determine $\frac{dC}{dt}$ if $C = 2\text{π}r$ and $\frac{dr}{dt} = 5$ at $r = 4$.
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$\frac{dC}{dt} = 10\text{π}$. Circumference formula differentiated.
$\frac{dC}{dt} = 10\text{π}$. Circumference formula differentiated.
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Identify the formula for the derivative of $\text{sin}(x)$.
Identify the formula for the derivative of $\text{sin}(x)$.
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$\frac{d}{dx}(\text{sin}(x)) = \text{cos}(x)$. Sine derivative is cosine.
$\frac{d}{dx}(\text{sin}(x)) = \text{cos}(x)$. Sine derivative is cosine.
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What is the relationship between rates of change in related rates problems?
What is the relationship between rates of change in related rates problems?
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They are connected by the chain rule. Chain rule connects changing variables.
They are connected by the chain rule. Chain rule connects changing variables.
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What is the chain rule for finding derivatives?
What is the chain rule for finding derivatives?
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$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Composes derivatives for nested functions.
$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Composes derivatives for nested functions.
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What is the derivative of $\frac{d}{dx}(\text{arcsin}(x))$?
What is the derivative of $\frac{d}{dx}(\text{arcsin}(x))$?
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$\frac{d}{dx}(\text{arcsin}(x)) = \frac{1}{\text{sqrt}(1-x^2)}$. Inverse sine derivative formula.
$\frac{d}{dx}(\text{arcsin}(x)) = \frac{1}{\text{sqrt}(1-x^2)}$. Inverse sine derivative formula.
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What is the derivative of $\frac{d}{dx}(\text{arctan}(x))$?
What is the derivative of $\frac{d}{dx}(\text{arctan}(x))$?
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$\frac{d}{dx}(\text{arctan}(x)) = \frac{1}{1+x^2}$. Inverse tangent derivative formula.
$\frac{d}{dx}(\text{arctan}(x)) = \frac{1}{1+x^2}$. Inverse tangent derivative formula.
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What is the derivative of $\frac{d}{dx}(\text{arccos}(x))$?
What is the derivative of $\frac{d}{dx}(\text{arccos}(x))$?
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$\frac{d}{dx}(\text{arccos}(x)) = -\frac{1}{\text{sqrt}(1-x^2)}$. Inverse cosine derivative is negative.
$\frac{d}{dx}(\text{arccos}(x)) = -\frac{1}{\text{sqrt}(1-x^2)}$. Inverse cosine derivative is negative.
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Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $r = 7$ and $\frac{dr}{dt} = 0.5$.
Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $r = 7$ and $\frac{dr}{dt} = 0.5$.
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$\frac{dA}{dt} = 7\pi$. Circle area: $\frac{dA}{dt} = 2\pi r \cdot 0.5 = 7\pi$.
$\frac{dA}{dt} = 7\pi$. Circle area: $\frac{dA}{dt} = 2\pi r \cdot 0.5 = 7\pi$.
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Determine $\frac{dC}{dt}$ for $C = 2\text{π}r$ if $r = 10$ and $\frac{dr}{dt} = 0.1$.
Determine $\frac{dC}{dt}$ for $C = 2\text{π}r$ if $r = 10$ and $\frac{dr}{dt} = 0.1$.
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$\frac{dC}{dt} = 0.2\text{π}$. Circumference: $\frac{dC}{dt} = 2\pi \cdot 0.1 = 0.2\pi$.
$\frac{dC}{dt} = 0.2\text{π}$. Circumference: $\frac{dC}{dt} = 2\pi \cdot 0.1 = 0.2\pi$.
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Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $\frac{dr}{dt} = 1$ and $r = 5$.
Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $\frac{dr}{dt} = 1$ and $r = 5$.
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$\frac{dA}{dt} = 10\pi$. Circle area formula differentiated.
$\frac{dA}{dt} = 10\pi$. Circle area formula differentiated.
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What is the basic strategy for solving related rates problems?
What is the basic strategy for solving related rates problems?
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Identify variables, relate them, differentiate, solve. Standard approach for related rates.
Identify variables, relate them, differentiate, solve. Standard approach for related rates.
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Calculate $\frac{dh}{dt}$ for $h = \text{tan}(t)$ when $t = \frac{\text{π}}{4}$.
Calculate $\frac{dh}{dt}$ for $h = \text{tan}(t)$ when $t = \frac{\text{π}}{4}$.
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$\frac{dh}{dt} = 2$. At $t = \frac{\pi}{4}$, $\sec^2(\frac{\pi}{4}) = 2$.
$\frac{dh}{dt} = 2$. At $t = \frac{\pi}{4}$, $\sec^2(\frac{\pi}{4}) = 2$.
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What is the derivative of $\frac{d}{dx}(\text{log}_a{x})$?
What is the derivative of $\frac{d}{dx}(\text{log}_a{x})$?
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$\frac{d}{dx}(\text{log}_a{x}) = \frac{1}{x\text{ln}(a)}$. Logarithm derivative with base $a$.
$\frac{d}{dx}(\text{log}_a{x}) = \frac{1}{x\text{ln}(a)}$. Logarithm derivative with base $a$.
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Determine $\frac{dA}{dt}$ for $A = s^2$ if $s = 10$ and $\frac{ds}{dt} = 2$.
Determine $\frac{dA}{dt}$ for $A = s^2$ if $s = 10$ and $\frac{ds}{dt} = 2$.
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$\frac{dA}{dt} = 40$. Square area formula: $\frac{dA}{dt} = 2s \cdot 2 = 40$.
$\frac{dA}{dt} = 40$. Square area formula: $\frac{dA}{dt} = 2s \cdot 2 = 40$.
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What is the derivative of $\frac{d}{dx}(\text{ln}(kx))$?
What is the derivative of $\frac{d}{dx}(\text{ln}(kx))$?
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$\frac{d}{dx}(\text{ln}(kx)) = \frac{1}{x}$. Chain rule cancels constant $k$.
$\frac{d}{dx}(\text{ln}(kx)) = \frac{1}{x}$. Chain rule cancels constant $k$.
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What is the derivative of $\frac{d}{dx}(\text{cot}(x))$?
What is the derivative of $\frac{d}{dx}(\text{cot}(x))$?
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$\frac{d}{dx}(\text{cot}(x)) = -\text{csc}^2(x)$. Cotangent derivative uses cosecant squared.
$\frac{d}{dx}(\text{cot}(x)) = -\text{csc}^2(x)$. Cotangent derivative uses cosecant squared.
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What is the derivative of $\frac{d}{dx}(\text{csc}(x))$?
What is the derivative of $\frac{d}{dx}(\text{csc}(x))$?
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$\frac{d}{dx}(\text{csc}(x)) = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative is negative.
$\frac{d}{dx}(\text{csc}(x)) = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative is negative.
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What is the formula for the derivative of $\frac{f}{g}$?
What is the formula for the derivative of $\frac{f}{g}$?
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$(f/g)' = \frac{f'g - fg'}{g^2}$. Quotient rule for derivatives.
$(f/g)' = \frac{f'g - fg'}{g^2}$. Quotient rule for derivatives.
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What is the derivative of $\frac{d}{dx}(\text{e}^{kx})$?
What is the derivative of $\frac{d}{dx}(\text{e}^{kx})$?
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$\frac{d}{dx}(\text{e}^{kx}) = k\text{e}^{kx}$. Chain rule with exponential function.
$\frac{d}{dx}(\text{e}^{kx}) = k\text{e}^{kx}$. Chain rule with exponential function.
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What is the derivative of $\frac{d}{dx}(\text{ln}(x))$?
What is the derivative of $\frac{d}{dx}(\text{ln}(x))$?
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$\frac{d}{dx}(\text{ln}(x)) = \frac{1}{x}$. Natural logarithm derivative formula.
$\frac{d}{dx}(\text{ln}(x)) = \frac{1}{x}$. Natural logarithm derivative formula.
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What is the derivative of $\frac{d}{dx}(\frac{1}{x})$?
What is the derivative of $\frac{d}{dx}(\frac{1}{x})$?
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$\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}$. Power rule with $n = -1$.
$\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}$. Power rule with $n = -1$.
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State the formula for the derivative of $\frac{d}{dx}(e^x)$.
State the formula for the derivative of $\frac{d}{dx}(e^x)$.
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$\frac{d}{dx}(e^x) = e^x$. Exponential function derivative equals itself.
$\frac{d}{dx}(e^x) = e^x$. Exponential function derivative equals itself.
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