Chain Rule - AP Calculus BC
Card 1 of 30
Identify $u$ in $h(x) = (3x^2 + 2x)^5$.
Identify $u$ in $h(x) = (3x^2 + 2x)^5$.
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$u = 3x^2 + 2x$. The expression being raised to the 5th power.
$u = 3x^2 + 2x$. The expression being raised to the 5th power.
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Find the derivative of $\text{sin}^2(x)$.
Find the derivative of $\text{sin}^2(x)$.
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$2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x) \times \cos(x)$ or $\sin(2x)$.
$2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x) \times \cos(x)$ or $\sin(2x)$.
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Differentiate $y = \text{sin}(\text{ln}(x^2))$.
Differentiate $y = \text{sin}(\text{ln}(x^2))$.
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$\frac{2x\text{cos}(\text{ln}(x^2))}{x^2}$. Chain rule twice: $\cos(\ln(x^2)) \times \frac{2x}{x^2}$.
$\frac{2x\text{cos}(\text{ln}(x^2))}{x^2}$. Chain rule twice: $\cos(\ln(x^2)) \times \frac{2x}{x^2}$.
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Differentiate $f(x) = \text{ln}(\text{e}^x + 1)$.
Differentiate $f(x) = \text{ln}(\text{e}^x + 1)$.
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$\frac{\text{e}^x}{\text{e}^x + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = e^x + 1$.
$\frac{\text{e}^x}{\text{e}^x + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = e^x + 1$.
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Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$.
Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$.
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$2x$. Simplifies to $x^2$ since $\ln$ and $e$ cancel.
$2x$. Simplifies to $x^2$ since $\ln$ and $e$ cancel.
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Identify $u$ in $y = (\text{ln}(x))^4$.
Identify $u$ in $y = (\text{ln}(x))^4$.
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$u = \text{ln}(x)$. The natural logarithm function raised to power 4.
$u = \text{ln}(x)$. The natural logarithm function raised to power 4.
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Differentiate $y = \text{e}^{\text{tan}(x)}$.
Differentiate $y = \text{e}^{\text{tan}(x)}$.
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$\text{sec}^2(x)\text{e}^{\text{tan}(x)}$. Chain rule: $e^u \times u'$ where $u = \tan(x)$.
$\text{sec}^2(x)\text{e}^{\text{tan}(x)}$. Chain rule: $e^u \times u'$ where $u = \tan(x)$.
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Which function is the inner function in $f(g(x)) = \text{e}^{3x+7}$?
Which function is the inner function in $f(g(x)) = \text{e}^{3x+7}$?
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$g(x) = 3x+7$. The linear expression in the exponent.
$g(x) = 3x+7$. The linear expression in the exponent.
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What is $y'$ if $y = \text{cos}(5x)$?
What is $y'$ if $y = \text{cos}(5x)$?
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$-5\text{sin}(5x)$. Chain rule: $-\sin(u) \times u'$ where $u = 5x$.
$-5\text{sin}(5x)$. Chain rule: $-\sin(u) \times u'$ where $u = 5x$.
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Differentiate $y = \text{e}^{3x+7}$.
Differentiate $y = \text{e}^{3x+7}$.
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$3\text{e}^{3x+7}$. Chain rule: $e^u \times u'$ where $u = 3x + 7$.
$3\text{e}^{3x+7}$. Chain rule: $e^u \times u'$ where $u = 3x + 7$.
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Differentiate $y = \text{e}^{\text{ln}(x)}$.
Differentiate $y = \text{e}^{\text{ln}(x)}$.
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$\frac{1}{x}$. Simplifies to $x$ since $e^{\ln(x)} = x$.
$\frac{1}{x}$. Simplifies to $x$ since $e^{\ln(x)} = x$.
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Differentiate $y = \text{cos}^{-1}(5x)$.
Differentiate $y = \text{cos}^{-1}(5x)$.
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$-\frac{5}{\text{sqrt}(1 - 25x^2)}$. Chain rule: $-\frac{1}{\sqrt{1-u^2}} \times u'$ where $u = 5x$.
$-\frac{5}{\text{sqrt}(1 - 25x^2)}$. Chain rule: $-\frac{1}{\sqrt{1-u^2}} \times u'$ where $u = 5x$.
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Find $f'(x)$ if $f(x) = \text{cos}(\text{e}^x)$.
Find $f'(x)$ if $f(x) = \text{cos}(\text{e}^x)$.
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$-\text{e}^x\text{sin}(\text{e}^x)$. Chain rule: $-\sin(u) \times u'$ where $u = e^x$.
$-\text{e}^x\text{sin}(\text{e}^x)$. Chain rule: $-\sin(u) \times u'$ where $u = e^x$.
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Find $d/dx$ of $y = \text{cos}(\text{e}^{2x})$.
Find $d/dx$ of $y = \text{cos}(\text{e}^{2x})$.
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$-2\text{e}^{2x}\text{sin}(\text{e}^{2x})$. Chain rule: $-\sin(u) \times u'$ where $u = e^{2x}$.
$-2\text{e}^{2x}\text{sin}(\text{e}^{2x})$. Chain rule: $-\sin(u) \times u'$ where $u = e^{2x}$.
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Differentiate $y = \text{tan}^2(3x)$ using the Chain Rule.
Differentiate $y = \text{tan}^2(3x)$ using the Chain Rule.
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$6\text{tan}(3x)\text{sec}^2(3x)$. Chain rule: $2u \times u'$ where $u = \tan(3x)$.
$6\text{tan}(3x)\text{sec}^2(3x)$. Chain rule: $2u \times u'$ where $u = \tan(3x)$.
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Differentiate $f(x) = \text{e}^{x^3}$.
Differentiate $f(x) = \text{e}^{x^3}$.
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$3x^2\text{e}^{x^3}$. Chain rule: $e^u \times u'$ where $u = x^3$.
$3x^2\text{e}^{x^3}$. Chain rule: $e^u \times u'$ where $u = x^3$.
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Differentiate $f(x) = \text{cos}^3(x)$.
Differentiate $f(x) = \text{cos}^3(x)$.
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$-3\text{cos}^2(x)\text{sin}(x)$. Chain rule: $3\cos^2(x) \times (-\sin(x))$.
$-3\text{cos}^2(x)\text{sin}(x)$. Chain rule: $3\cos^2(x) \times (-\sin(x))$.
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Which function is the outer function in $f(g(x)) = \text{e}^{3x+7}$?
Which function is the outer function in $f(g(x)) = \text{e}^{3x+7}$?
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$f(u) = \text{e}^u$. The exponential function wraps the linear expression.
$f(u) = \text{e}^u$. The exponential function wraps the linear expression.
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Evaluate $\frac{d}{dx}[\text{ln}(x^2 + 1)]$.
Evaluate $\frac{d}{dx}[\text{ln}(x^2 + 1)]$.
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$\frac{2x}{x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = x^2 + 1$.
$\frac{2x}{x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = x^2 + 1$.
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Identify the outer function in $f(g(x)) = \tan(\frac{x^2}{3})$.
Identify the outer function in $f(g(x)) = \tan(\frac{x^2}{3})$.
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$f(u) = \tan(u)$. The tangent function wraps the inner expression.
$f(u) = \tan(u)$. The tangent function wraps the inner expression.
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Differentiate $y = (2x + 5)^6$.
Differentiate $y = (2x + 5)^6$.
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$12(2x+5)^5$. Power rule with chain rule: $6u^5 \times u'$.
$12(2x+5)^5$. Power rule with chain rule: $6u^5 \times u'$.
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Differentiate $g(x) = \text{ln}(4x^2 + 1)$.
Differentiate $g(x) = \text{ln}(4x^2 + 1)$.
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$\frac{8x}{4x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = 4x^2 + 1$.
$\frac{8x}{4x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = 4x^2 + 1$.
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Differentiate $e^{x^2}$ using the Chain Rule.
Differentiate $e^{x^2}$ using the Chain Rule.
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$2xe^{x^2}$. Chain rule: derivative of $e^u$ is $e^u$ times $u'$.
$2xe^{x^2}$. Chain rule: derivative of $e^u$ is $e^u$ times $u'$.
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Identify the inner function in $f(g(x)) = \tan(\frac{x^2}{3})$.
Identify the inner function in $f(g(x)) = \tan(\frac{x^2}{3})$.
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$g(x) = \frac{x^2}{3}$. The expression inside the tangent function.
$g(x) = \frac{x^2}{3}$. The expression inside the tangent function.
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State the formula for the Chain Rule.
State the formula for the Chain Rule.
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$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Fundamental chain rule formula for composite functions.
$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Fundamental chain rule formula for composite functions.
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Find $y'$ if $y = \text{tan}(\text{ln}(x))$.
Find $y'$ if $y = \text{tan}(\text{ln}(x))$.
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$\frac{1}{x\text{cos}^2(\text{ln}(x))}$. Chain rule twice: $\sec^2(\ln(x)) \times \frac{1}{x}$.
$\frac{1}{x\text{cos}^2(\text{ln}(x))}$. Chain rule twice: $\sec^2(\ln(x)) \times \frac{1}{x}$.
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Differentiate $y = (\text{ln}(x))^4$ using the Chain Rule.
Differentiate $y = (\text{ln}(x))^4$ using the Chain Rule.
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$\frac{4(\text{ln}(x))^3}{x}$. Power rule with chain rule: $4u^3 \times u'$.
$\frac{4(\text{ln}(x))^3}{x}$. Power rule with chain rule: $4u^3 \times u'$.
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Find $f'(x)$ if $f(x) = \text{sin}(x^3)$.
Find $f'(x)$ if $f(x) = \text{sin}(x^3)$.
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$3x^2 \text{cos}(x^3)$. Chain rule: $\cos(u) \times u'$ where $u = x^3$.
$3x^2 \text{cos}(x^3)$. Chain rule: $\cos(u) \times u'$ where $u = x^3$.
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Differentiate $f(x) = \text{e}^{\text{sin}(x)}$.
Differentiate $f(x) = \text{e}^{\text{sin}(x)}$.
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$\text{cos}(x)\text{e}^{\text{sin}(x)}$. Chain rule: $e^u \times u'$ where $u = \sin(x)$.
$\text{cos}(x)\text{e}^{\text{sin}(x)}$. Chain rule: $e^u \times u'$ where $u = \sin(x)$.
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What is the derivative of $f(g(x))$ using the Chain Rule?
What is the derivative of $f(g(x))$ using the Chain Rule?
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$f'(g(x)) \times g'(x)$. Chain rule applied to general composite function $f(g(x))$.
$f'(g(x)) \times g'(x)$. Chain rule applied to general composite function $f(g(x))$.
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