Differentiating Inverse Trigonometric Functions - AP Calculus BC
Card 1 of 30
Evaluate $\frac{d}{dx}(\tan^{-1}(x))$ at $x = 1$.
Evaluate $\frac{d}{dx}(\tan^{-1}(x))$ at $x = 1$.
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$\frac{1}{2}$. Substitute $x = 1$ into $\frac{1}{1+x^2}$.
$\frac{1}{2}$. Substitute $x = 1$ into $\frac{1}{1+x^2}$.
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What is the chain rule application for $y = \cot^{-1}(g(x))$?
What is the chain rule application for $y = \cot^{-1}(g(x))$?
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$-\frac{g'(x)}{1+(g(x))^2}$. General chain rule pattern for inverse cotangent composition.
$-\frac{g'(x)}{1+(g(x))^2}$. General chain rule pattern for inverse cotangent composition.
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Identify the derivative of $y = \cot^{-1}(\sqrt{x})$.
Identify the derivative of $y = \cot^{-1}(\sqrt{x})$.
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$-\frac{1}{2\sqrt{x}(1+x)}$. Chain rule with $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$ applied to $\cot^{-1}$.
$-\frac{1}{2\sqrt{x}(1+x)}$. Chain rule with $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$ applied to $\cot^{-1}$.
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Identify the derivative of $y = \cos^{-1}(x^3)$.
Identify the derivative of $y = \cos^{-1}(x^3)$.
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$-\frac{3x^2}{\sqrt{1-x^6}}$. Chain rule with $\frac{d}{dx}(x^3) = 3x^2$ applied to $\cos^{-1}$ formula.
$-\frac{3x^2}{\sqrt{1-x^6}}$. Chain rule with $\frac{d}{dx}(x^3) = 3x^2$ applied to $\cos^{-1}$ formula.
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Evaluate $\frac{d}{dx}(\sin^{-1}(x))$ at $x = \frac{1}{2}$.
Evaluate $\frac{d}{dx}(\sin^{-1}(x))$ at $x = \frac{1}{2}$.
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$\frac{2\sqrt{3}}{3}$. Substitute $x = \frac{1}{2}$ into $\frac{1}{\sqrt{1-x^2}}$.
$\frac{2\sqrt{3}}{3}$. Substitute $x = \frac{1}{2}$ into $\frac{1}{\sqrt{1-x^2}}$.
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Find $\frac{d}{dx}(\cot^{-1}(\sqrt{2x}))$.
Find $\frac{d}{dx}(\cot^{-1}(\sqrt{2x}))$.
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$-\frac{1}{2\sqrt{2x}(1+2x)}$. Chain rule with $\frac{d}{dx}(\sqrt{2x}) = \frac{1}{\sqrt{2x}}$ applied to $\cot^{-1}$.
$-\frac{1}{2\sqrt{2x}(1+2x)}$. Chain rule with $\frac{d}{dx}(\sqrt{2x}) = \frac{1}{\sqrt{2x}}$ applied to $\cot^{-1}$.
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State the derivative of $\sec^{-1}(x)$.
State the derivative of $\sec^{-1}(x)$.
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$\frac{1}{|x|\sqrt{x^2-1}}$. Basic derivative formula for inverse secant function.
$\frac{1}{|x|\sqrt{x^2-1}}$. Basic derivative formula for inverse secant function.
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State the derivative of $\tan^{-1}(x)$.
State the derivative of $\tan^{-1}(x)$.
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$\frac{1}{1+x^2}$. Basic derivative formula for inverse tangent function.
$\frac{1}{1+x^2}$. Basic derivative formula for inverse tangent function.
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Identify the derivative of $y = \csc^{-1}(x^5)$.
Identify the derivative of $y = \csc^{-1}(x^5)$.
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$-\frac{5x^4}{|x^5|\sqrt{x^{10}-1}}$. Chain rule with $\frac{d}{dx}(x^5) = 5x^4$ applied to $\csc^{-1}$ formula.
$-\frac{5x^4}{|x^5|\sqrt{x^{10}-1}}$. Chain rule with $\frac{d}{dx}(x^5) = 5x^4$ applied to $\csc^{-1}$ formula.
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What is the chain rule application for $y = \sin^{-1}(g(x))$?
What is the chain rule application for $y = \sin^{-1}(g(x))$?
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$\frac{g'(x)}{\sqrt{1-(g(x))^2}}$. General chain rule pattern for inverse sine composition.
$\frac{g'(x)}{\sqrt{1-(g(x))^2}}$. General chain rule pattern for inverse sine composition.
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Find $\frac{d}{dx}(\sin^{-1}(2x))$.
Find $\frac{d}{dx}(\sin^{-1}(2x))$.
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$\frac{2}{\sqrt{1-(2x)^2}}$. Chain rule with $\frac{d}{dx}(2x) = 2$ applied to $\sin^{-1}$ formula.
$\frac{2}{\sqrt{1-(2x)^2}}$. Chain rule with $\frac{d}{dx}(2x) = 2$ applied to $\sin^{-1}$ formula.
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What is the chain rule application for $y = \csc^{-1}(g(x))$?
What is the chain rule application for $y = \csc^{-1}(g(x))$?
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$-\frac{g'(x)}{|g(x)|\sqrt{(g(x))^2-1}}$. General chain rule pattern for inverse cosecant composition.
$-\frac{g'(x)}{|g(x)|\sqrt{(g(x))^2-1}}$. General chain rule pattern for inverse cosecant composition.
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State the derivative of $\csc^{-1}(x)$.
State the derivative of $\csc^{-1}(x)$.
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$-\frac{1}{|x|\sqrt{x^2-1}}$. Basic derivative formula for inverse cosecant function.
$-\frac{1}{|x|\sqrt{x^2-1}}$. Basic derivative formula for inverse cosecant function.
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Find $\frac{d}{dx}(\tan^{-1}(\sqrt{3x}))$.
Find $\frac{d}{dx}(\tan^{-1}(\sqrt{3x}))$.
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$\frac{3}{2\sqrt{3x}(1+3x)}$. Chain rule with $\frac{d}{dx}(\sqrt{3x}) = \frac{3}{2\sqrt{3x}}$ applied to $\tan^{-1}$.
$\frac{3}{2\sqrt{3x}(1+3x)}$. Chain rule with $\frac{d}{dx}(\sqrt{3x}) = \frac{3}{2\sqrt{3x}}$ applied to $\tan^{-1}$.
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Find $\frac{d}{dx}(\csc^{-1}(2x^3))$.
Find $\frac{d}{dx}(\csc^{-1}(2x^3))$.
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$-\frac{6x^2}{|2x^3|\sqrt{(2x^3)^2-1}}$. Chain rule with $\frac{d}{dx}(2x^3) = 6x^2$ applied to $\csc^{-1}$.
$-\frac{6x^2}{|2x^3|\sqrt{(2x^3)^2-1}}$. Chain rule with $\frac{d}{dx}(2x^3) = 6x^2$ applied to $\csc^{-1}$.
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Find $\frac{d}{dx}(\sec^{-1}(5x^2))$.
Find $\frac{d}{dx}(\sec^{-1}(5x^2))$.
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$\frac{10x}{|5x^2|\sqrt{(5x^2)^2-1}}$. Chain rule with $\frac{d}{dx}(5x^2) = 10x$ applied to $\sec^{-1}$.
$\frac{10x}{|5x^2|\sqrt{(5x^2)^2-1}}$. Chain rule with $\frac{d}{dx}(5x^2) = 10x$ applied to $\sec^{-1}$.
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Find $\frac{d}{dx}(\cot^{-1}(4x))$.
Find $\frac{d}{dx}(\cot^{-1}(4x))$.
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$-\frac{4}{1+(4x)^2}$. Chain rule with $\frac{d}{dx}(4x) = 4$ applied to $\cot^{-1}$ formula.
$-\frac{4}{1+(4x)^2}$. Chain rule with $\frac{d}{dx}(4x) = 4$ applied to $\cot^{-1}$ formula.
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State the derivative of $\cos^{-1}(x)$.
State the derivative of $\cos^{-1}(x)$.
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$-\frac{1}{\sqrt{1-x^2}}$. Basic derivative formula for inverse cosine function.
$-\frac{1}{\sqrt{1-x^2}}$. Basic derivative formula for inverse cosine function.
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Find $\frac{d}{dx}(x \cos^{-1}(x))$.
Find $\frac{d}{dx}(x \cos^{-1}(x))$.
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$\cos^{-1}(x) - \frac{x}{\sqrt{1-x^2}}$. Product rule: $(uv)' = u'v + uv'$ applied to $x\cos^{-1}(x)$.
$\cos^{-1}(x) - \frac{x}{\sqrt{1-x^2}}$. Product rule: $(uv)' = u'v + uv'$ applied to $x\cos^{-1}(x)$.
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Evaluate $\frac{d}{dx}(\csc^{-1}(x))$ at $x = 2$.
Evaluate $\frac{d}{dx}(\csc^{-1}(x))$ at $x = 2$.
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$-\frac{1}{2\sqrt{3}}$. Substitute $x = 2$ into $-\frac{1}{|x|\sqrt{x^2-1}}$.
$-\frac{1}{2\sqrt{3}}$. Substitute $x = 2$ into $-\frac{1}{|x|\sqrt{x^2-1}}$.
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Evaluate $\frac{d}{dx}(\sec^{-1}(x))$ at $x = 2$.
Evaluate $\frac{d}{dx}(\sec^{-1}(x))$ at $x = 2$.
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$\frac{1}{2\sqrt{3}}$. Substitute $x = 2$ into $\frac{1}{|x|\sqrt{x^2-1}}$.
$\frac{1}{2\sqrt{3}}$. Substitute $x = 2$ into $\frac{1}{|x|\sqrt{x^2-1}}$.
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Find $\frac{d}{dx}(\sin^{-1}(3x^2))$.
Find $\frac{d}{dx}(\sin^{-1}(3x^2))$.
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$\frac{6x}{\sqrt{1-(3x^2)^2}}$. Chain rule with $\frac{d}{dx}(3x^2) = 6x$ applied to $\sin^{-1}$.
$\frac{6x}{\sqrt{1-(3x^2)^2}}$. Chain rule with $\frac{d}{dx}(3x^2) = 6x$ applied to $\sin^{-1}$.
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What is the chain rule application for $y = \tan^{-1}(g(x))$?
What is the chain rule application for $y = \tan^{-1}(g(x))$?
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$\frac{g'(x)}{1+(g(x))^2}$. General chain rule pattern for inverse tangent composition.
$\frac{g'(x)}{1+(g(x))^2}$. General chain rule pattern for inverse tangent composition.
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Find $\frac{d}{dx}(\tan^{-1}(5x))$.
Find $\frac{d}{dx}(\tan^{-1}(5x))$.
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$\frac{5}{1+(5x)^2}$. Chain rule with $\frac{d}{dx}(5x) = 5$ applied to $\tan^{-1}$ formula.
$\frac{5}{1+(5x)^2}$. Chain rule with $\frac{d}{dx}(5x) = 5$ applied to $\tan^{-1}$ formula.
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What is the chain rule application for $y = \sec^{-1}(g(x))$?
What is the chain rule application for $y = \sec^{-1}(g(x))$?
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$\frac{g'(x)}{|g(x)|\sqrt{(g(x))^2-1}}$. General chain rule pattern for inverse secant composition.
$\frac{g'(x)}{|g(x)|\sqrt{(g(x))^2-1}}$. General chain rule pattern for inverse secant composition.
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Identify the derivative of $y = \sin^{-1}(x^2)$.
Identify the derivative of $y = \sin^{-1}(x^2)$.
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$\frac{2x}{\sqrt{1-x^4}}$. Chain rule with $\frac{d}{dx}(x^2) = 2x$ applied to $\sin^{-1}$ formula.
$\frac{2x}{\sqrt{1-x^4}}$. Chain rule with $\frac{d}{dx}(x^2) = 2x$ applied to $\sin^{-1}$ formula.
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Find $\frac{d}{dx}(\csc^{-1}(7x))$.
Find $\frac{d}{dx}(\csc^{-1}(7x))$.
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$-\frac{7}{|7x|\sqrt{(7x)^2-1}}$. Chain rule with $\frac{d}{dx}(7x) = 7$ applied to $\csc^{-1}$ formula.
$-\frac{7}{|7x|\sqrt{(7x)^2-1}}$. Chain rule with $\frac{d}{dx}(7x) = 7$ applied to $\csc^{-1}$ formula.
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Find $\frac{d}{dx}(\sec^{-1}(6x))$.
Find $\frac{d}{dx}(\sec^{-1}(6x))$.
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$\frac{6}{|6x|\sqrt{(6x)^2-1}}$. Chain rule with $\frac{d}{dx}(6x) = 6$ applied to $\sec^{-1}$ formula.
$\frac{6}{|6x|\sqrt{(6x)^2-1}}$. Chain rule with $\frac{d}{dx}(6x) = 6$ applied to $\sec^{-1}$ formula.
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State the derivative of $\sin^{-1}(x)$.
State the derivative of $\sin^{-1}(x)$.
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$\frac{1}{\sqrt{1-x^2}}$. Basic derivative formula for inverse sine function.
$\frac{1}{\sqrt{1-x^2}}$. Basic derivative formula for inverse sine function.
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Find $\frac{d}{dx}(\cos^{-1}(3x))$.
Find $\frac{d}{dx}(\cos^{-1}(3x))$.
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$-\frac{3}{\sqrt{1-(3x)^2}}$. Chain rule with $\frac{d}{dx}(3x) = 3$ applied to $\cos^{-1}$ formula.
$-\frac{3}{\sqrt{1-(3x)^2}}$. Chain rule with $\frac{d}{dx}(3x) = 3$ applied to $\cos^{-1}$ formula.
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