Chain Rule - AP Calculus AB
Card 1 of 30
Differentiate $y = \text{cos}(\text{ln}(x^2))$.
Differentiate $y = \text{cos}(\text{ln}(x^2))$.
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$ -\frac{2\text{sin}(\text{ln}(x^2))}{x} $. Cosine composition with natural log of $x^2$.
$ -\frac{2\text{sin}(\text{ln}(x^2))}{x} $. Cosine composition with natural log of $x^2$.
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Differentiate $y = \sin^3(x)$.
Differentiate $y = \sin^3(x)$.
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$3\sin^2(x)\cos(x)$. Power rule: $3\sin^2(x) \times \cos(x)$.
$3\sin^2(x)\cos(x)$. Power rule: $3\sin^2(x) \times \cos(x)$.
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Differentiate $y = e^{\tan(x)}$
Differentiate $y = e^{\tan(x)}$
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$\sec^2(x) e^{\tan(x)}$. Exponential times derivative of tangent function.
$\sec^2(x) e^{\tan(x)}$. Exponential times derivative of tangent function.
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Differentiate $y = \text{cos}(5x)$.
Differentiate $y = \text{cos}(5x)$.
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$-5\text{sin}(5x)$. Derivative of cosine is $-\sin$ times inner derivative.
$-5\text{sin}(5x)$. Derivative of cosine is $-\sin$ times inner derivative.
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Find the derivative of $y = \frac{1}{\text{e}^{2x}}$.
Find the derivative of $y = \frac{1}{\text{e}^{2x}}$.
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$-2\text{e}^{-2x}$. Rewrite as $\text{e}^{-2x}$ and differentiate.
$-2\text{e}^{-2x}$. Rewrite as $\text{e}^{-2x}$ and differentiate.
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Differentiate $y = \text{ln}(\text{e}^{2x})$.
Differentiate $y = \text{ln}(\text{e}^{2x})$.
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- Simplifies to $\ln(\text{e}^{2x}) = 2x$.
- Simplifies to $\ln(\text{e}^{2x}) = 2x$.
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Differentiate $y = \text{e}^{\text{e}^x}$.
Differentiate $y = \text{e}^{\text{e}^x}$.
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$\text{e}^x\text{e}^{\text{e}^x}$. Double exponential: $\text{e}^x \times \text{e}^{\text{e}^x}$.
$\text{e}^x\text{e}^{\text{e}^x}$. Double exponential: $\text{e}^x \times \text{e}^{\text{e}^x}$.
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Differentiate $y = \text{tan}^3(x)$.
Differentiate $y = \text{tan}^3(x)$.
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$3\text{tan}^2(x)\text{sec}^2(x)$. Power rule: $3\tan^2(x) \times \sec^2(x)$.
$3\text{tan}^2(x)\text{sec}^2(x)$. Power rule: $3\tan^2(x) \times \sec^2(x)$.
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Differentiate $y = (5x + 3)^7$.
Differentiate $y = (5x + 3)^7$.
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$35(5x + 3)^6$. Power rule: $7(5x + 3)^6 \times 5$.
$35(5x + 3)^6$. Power rule: $7(5x + 3)^6 \times 5$.
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Differentiate $y = \text{cos}(\text{sin}(x))$.
Differentiate $y = \text{cos}(\text{sin}(x))$.
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$-\text{sin}(\text{sin}(x))\text{cos}(x)$. Cosine composition: $-\sin(\sin(x)) \times \cos(x)$.
$-\text{sin}(\text{sin}(x))\text{cos}(x)$. Cosine composition: $-\sin(\sin(x)) \times \cos(x)$.
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Differentiate $y = \text{sin}(\text{e}^{3x})$.
Differentiate $y = \text{sin}(\text{e}^{3x})$.
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$3\text{e}^{3x}\text{cos}(\text{e}^{3x})$. Sine composition: $\cos(\text{e}^{3x}) \times 3\text{e}^{3x}$.
$3\text{e}^{3x}\text{cos}(\text{e}^{3x})$. Sine composition: $\cos(\text{e}^{3x}) \times 3\text{e}^{3x}$.
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Differentiate $y = \text{tan}(\text{ln}(x))$.
Differentiate $y = \text{tan}(\text{ln}(x))$.
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$\frac{\text{sec}^2(\text{ln}(x))}{x}$. Tangent of natural log: $\sec^2(\ln(x)) \times \frac{1}{x}$.
$\frac{\text{sec}^2(\text{ln}(x))}{x}$. Tangent of natural log: $\sec^2(\ln(x)) \times \frac{1}{x}$.
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Differentiate $y = \text{cos}(\text{ln}(x))$.
Differentiate $y = \text{cos}(\text{ln}(x))$.
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$-\frac{\text{sin}(\text{ln}(x))}{x}$. Chain rule: $-\sin(\ln(x)) \times \frac{1}{x}$.
$-\frac{\text{sin}(\text{ln}(x))}{x}$. Chain rule: $-\sin(\ln(x)) \times \frac{1}{x}$.
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Differentiate $y = \text{ln}(7x^2 + 5)$.
Differentiate $y = \text{ln}(7x^2 + 5)$.
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$\frac{14x}{7x^2 + 5}$. Derivative of $\ln(u)$ is $\frac{1}{u} \times u'$.
$\frac{14x}{7x^2 + 5}$. Derivative of $\ln(u)$ is $\frac{1}{u} \times u'$.
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Differentiate $y = (\text{ln}(x))^3$.
Differentiate $y = (\text{ln}(x))^3$.
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$\frac{3(\text{ln}(x))^2}{x}$. Power rule: $3(\ln(x))^2 \times \frac{1}{x}$.
$\frac{3(\text{ln}(x))^2}{x}$. Power rule: $3(\ln(x))^2 \times \frac{1}{x}$.
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Differentiate $y = (\text{e}^x)^2$.
Differentiate $y = (\text{e}^x)^2$.
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$2\text{e}^{2x}$. Use power rule: $(\text{e}^x)^2 = \text{e}^{2x}$.
$2\text{e}^{2x}$. Use power rule: $(\text{e}^x)^2 = \text{e}^{2x}$.
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Differentiate $y = \frac{1}{\text{sin}(x^2)}$.
Differentiate $y = \frac{1}{\text{sin}(x^2)}$.
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$-\frac{2x\text{cos}(x^2)}{\text{sin}^2(x^2)}$. Use quotient rule or rewrite as $\csc(x^2)$.
$-\frac{2x\text{cos}(x^2)}{\text{sin}^2(x^2)}$. Use quotient rule or rewrite as $\csc(x^2)$.
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Differentiate $y = \text{cos}(2x^2)$.
Differentiate $y = \text{cos}(2x^2)$.
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$-4x\text{sin}(2x^2)$. Cosine derivative: $-\sin(2x^2) \times 4x$.
$-4x\text{sin}(2x^2)$. Cosine derivative: $-\sin(2x^2) \times 4x$.
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Differentiate $y = \text{sin}(\text{e}^x)$.
Differentiate $y = \text{sin}(\text{e}^x)$.
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$\text{e}^x\text{cos}(\text{e}^x)$. Cosine of $\text{e}^x$ times derivative of $\text{e}^x$.
$\text{e}^x\text{cos}(\text{e}^x)$. Cosine of $\text{e}^x$ times derivative of $\text{e}^x$.
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Identify the inner function in $y = (4x^2 + 1)^5$.
Identify the inner function in $y = (4x^2 + 1)^5$.
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$u = 4x^2 + 1$. The expression inside the power is the inner function.
$u = 4x^2 + 1$. The expression inside the power is the inner function.
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What is the derivative of $y = (3x + 2)^4$ using the Chain Rule?
What is the derivative of $y = (3x + 2)^4$ using the Chain Rule?
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$12(3x + 2)^3$. Power rule with chain rule: $4(3x + 2)^3 \times 3$.
$12(3x + 2)^3$. Power rule with chain rule: $4(3x + 2)^3 \times 3$.
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Differentiate $y = \text{tan}^2(x)$.
Differentiate $y = \text{tan}^2(x)$.
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$2\text{tan}(x)\text{sec}^2(x)$. Power rule: $2\tan(x) \times \sec^2(x)$.
$2\text{tan}(x)\text{sec}^2(x)$. Power rule: $2\tan(x) \times \sec^2(x)$.
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State the formula for the Chain Rule.
State the formula for the Chain Rule.
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$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Multiplies outer derivative by inner derivative.
$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Multiplies outer derivative by inner derivative.
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Differentiate $y = \text{ln}(x^2 + 2x)$.
Differentiate $y = \text{ln}(x^2 + 2x)$.
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$\frac{2x + 2}{x^2 + 2x}$. Natural log derivative: $\frac{1}{x^2 + 2x} \times (2x + 2)$.
$\frac{2x + 2}{x^2 + 2x}$. Natural log derivative: $\frac{1}{x^2 + 2x} \times (2x + 2)$.
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Identify the inner function in $y = \text{ln}(\text{cos}(x))$.
Identify the inner function in $y = \text{ln}(\text{cos}(x))$.
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$u = \text{cos}(x)$. The cosine function is inside the natural log.
$u = \text{cos}(x)$. The cosine function is inside the natural log.
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Differentiate $y = \text{e}^{\text{cos}(x)}$
Differentiate $y = \text{e}^{\text{cos}(x)}$
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$ -\text{sin}(x)\text{e}^{\text{cos}(x)} $. Exponential times derivative of exponent: $-\sin(x)$
$ -\text{sin}(x)\text{e}^{\text{cos}(x)} $. Exponential times derivative of exponent: $-\sin(x)$
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Differentiate $y = \text{sin}(\text{cos}(x))$.
Differentiate $y = \text{sin}(\text{cos}(x))$.
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$-\text{cos}(\text{cos}(x))\text{sin}(x)$. Composition: sine of cosine times derivative of cosine.
$-\text{cos}(\text{cos}(x))\text{sin}(x)$. Composition: sine of cosine times derivative of cosine.
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Differentiate $y = \tan(4x)$.
Differentiate $y = \tan(4x)$.
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$4\sec^2(4x)$. Tangent derivative is $\sec^2$ times inner derivative.
$4\sec^2(4x)$. Tangent derivative is $\sec^2$ times inner derivative.
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Find $\frac{d}{dx}(\text{e}^{3x+1})$.
Find $\frac{d}{dx}(\text{e}^{3x+1})$.
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$3\text{e}^{3x+1}$. Exponential function times derivative of exponent.
$3\text{e}^{3x+1}$. Exponential function times derivative of exponent.
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Differentiate $y = \text{e}^{x^2 + 3}$.
Differentiate $y = \text{e}^{x^2 + 3}$.
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$2x\text{e}^{x^2 + 3}$. Exponential function times derivative of exponent.
$2x\text{e}^{x^2 + 3}$. Exponential function times derivative of exponent.
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