Differentiating Inverse Trigonometric Functions - AP Calculus AB
Card 1 of 30
What is the derivative of $\text{arcsec}(x)$ at $x = 2$?
What is the derivative of $\text{arcsec}(x)$ at $x = 2$?
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$\frac{1}{2\sqrt{3}}$. Substitute $x = 2$ into $\frac{1}{|x|\sqrt{x^2-1}}$ formula.
$\frac{1}{2\sqrt{3}}$. Substitute $x = 2$ into $\frac{1}{|x|\sqrt{x^2-1}}$ formula.
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Identify the derivative of $\text{arcsin}(2x)$ with respect to $x$.
Identify the derivative of $\text{arcsin}(2x)$ with respect to $x$.
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$\frac{2}{\sqrt{1-4x^2)}$. Apply chain rule: derivative of $2x$ times $\frac{1}{\sqrt{1-(2x)^2}}$.
$\frac{2}{\sqrt{1-4x^2)}$. Apply chain rule: derivative of $2x$ times $\frac{1}{\sqrt{1-(2x)^2}}$.
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Identify the derivative of $\text{arccsc}(7x)$ with respect to $x$.
Identify the derivative of $\text{arccsc}(7x)$ with respect to $x$.
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$-\frac{7}{|7x|\text{sqrt}(49x^2-1)}$. Apply chain rule with arccsc derivative formula.
$-\frac{7}{|7x|\text{sqrt}(49x^2-1)}$. Apply chain rule with arccsc derivative formula.
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Evaluate the derivative of $y = \text{arccos}(\frac{x}{3})$ at $x = 1$.
Evaluate the derivative of $y = \text{arccos}(\frac{x}{3})$ at $x = 1$.
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$ -\frac{1}{\sqrt{8}} $. Use chain rule with $\frac{d}{dx}[\frac{x}{3}] = \frac{1}{3}$ and evaluate at $x = 1$.
$ -\frac{1}{\sqrt{8}} $. Use chain rule with $\frac{d}{dx}[\frac{x}{3}] = \frac{1}{3}$ and evaluate at $x = 1$.
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Determine the derivative of $y = \text{arcsec}(\text{sec}(x))$.
Determine the derivative of $y = \text{arcsec}(\text{sec}(x))$.
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$1$. Inverse function cancels on principal domain.
$1$. Inverse function cancels on principal domain.
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Determine the derivative of $y = \text{arcsin}(\text{sin}(x))$.
Determine the derivative of $y = \text{arcsin}(\text{sin}(x))$.
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$1$. Inverse function cancels on principal domain.
$1$. Inverse function cancels on principal domain.
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Evaluate the derivative of $y = \text{arcsin}(\frac{x}{2})$ at $x = 1$.
Evaluate the derivative of $y = \text{arcsin}(\frac{x}{2})$ at $x = 1$.
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$\frac{1}{\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[\frac{x}{2}] = \frac{1}{2}$ and evaluate at $x = 1$.
$\frac{1}{\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[\frac{x}{2}] = \frac{1}{2}$ and evaluate at $x = 1$.
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State the derivative of $\text{arcsec}(x)$.
State the derivative of $\text{arcsec}(x)$.
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$\frac{1}{|x| \sqrt{x^2 - 1}}$. Standard derivative formula for inverse secant function.
$\frac{1}{|x| \sqrt{x^2 - 1}}$. Standard derivative formula for inverse secant function.
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State the derivative of $\text{arctan}(x)$.
State the derivative of $\text{arctan}(x)$.
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$\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.
$\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.
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What is the derivative of $\text{arctan}(x)$ at $x = 1$?
What is the derivative of $\text{arctan}(x)$ at $x = 1$?
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$\frac{1}{2}$. Substitute $x = 1$ into $\frac{1}{1+x^2}$ formula.
$\frac{1}{2}$. Substitute $x = 1$ into $\frac{1}{1+x^2}$ formula.
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Evaluate the derivative of $y = \text{arctan}(2x)$ at $x = 0.5$.
Evaluate the derivative of $y = \text{arctan}(2x)$ at $x = 0.5$.
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$\frac{2}{5}$. Use chain rule with $\frac{d}{dx}[2x] = 2$ and evaluate at $x = 0.5$.
$\frac{2}{5}$. Use chain rule with $\frac{d}{dx}[2x] = 2$ and evaluate at $x = 0.5$.
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What is the derivative of $\text{arcsin}(x)$ at $x = 0$?
What is the derivative of $\text{arcsin}(x)$ at $x = 0$?
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$1$. Substitute $x = 0$ into $\frac{1}{\sqrt{1-x^2}}$ formula.
$1$. Substitute $x = 0$ into $\frac{1}{\sqrt{1-x^2}}$ formula.
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Determine the derivative of $y = \text{arccot}(\text{cot}(x))$.
Determine the derivative of $y = \text{arccot}(\text{cot}(x))$.
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$-1$. Inverse function cancels on principal domain.
$-1$. Inverse function cancels on principal domain.
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Determine the derivative of $y = \text{arctan}(\text{tan}(x))$.
Determine the derivative of $y = \text{arctan}(\text{tan}(x))$.
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$1$. Inverse function cancels on principal domain.
$1$. Inverse function cancels on principal domain.
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Find the derivative of $y = \text{arccot}(\frac{1}{x})$ at $x = 1$.
Find the derivative of $y = \text{arccot}(\frac{1}{x})$ at $x = 1$.
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$-1$. Use chain rule with $\frac{d}{dx}[\frac{1}{x}] = -\frac{1}{x^2}$ and evaluate at $x = 1$.
$-1$. Use chain rule with $\frac{d}{dx}[\frac{1}{x}] = -\frac{1}{x^2}$ and evaluate at $x = 1$.
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State the derivative of $\text{arccot}(x)$.
State the derivative of $\text{arccot}(x)$.
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$-\frac{1}{1+x^2}$. Standard derivative formula for inverse cotangent function.
$-\frac{1}{1+x^2}$. Standard derivative formula for inverse cotangent function.
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Evaluate the derivative of $\text{arccot}(x)$ at $x = 1$.
Evaluate the derivative of $\text{arccot}(x)$ at $x = 1$.
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$ -\frac{1}{2} $. Substitute $x = 1$ into $ -\frac{1}{1+x^2} $ formula.
$ -\frac{1}{2} $. Substitute $x = 1$ into $ -\frac{1}{1+x^2} $ formula.
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Identify the derivative of $\text{arccot}(8x)$ with respect to $x$.
Identify the derivative of $\text{arccot}(8x)$ with respect to $x$.
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$-\frac{8}{1+64x^2}$. Apply chain rule: derivative of $8x$ times $-\frac{1}{1+(8x)^2}$.
$-\frac{8}{1+64x^2}$. Apply chain rule: derivative of $8x$ times $-\frac{1}{1+(8x)^2}$.
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Evaluate the derivative of $\text{arccsc}(x)$ at $x = -2$.
Evaluate the derivative of $\text{arccsc}(x)$ at $x = -2$.
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$\frac{1}{2\text{sqrt}(3)}$. Substitute $x = -2$ into $-\frac{1}{|x|\sqrt{x^2-1}}$ formula.
$\frac{1}{2\text{sqrt}(3)}$. Substitute $x = -2$ into $-\frac{1}{|x|\sqrt{x^2-1}}$ formula.
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Identify the derivative of $\text{arctan}(5x)$ with respect to $x$.
Identify the derivative of $\text{arctan}(5x)$ with respect to $x$.
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$\frac{5}{1+25x^2}$. Apply chain rule: derivative of $5x$ times $\frac{1}{1+(5x)^2}$.
$\frac{5}{1+25x^2}$. Apply chain rule: derivative of $5x$ times $\frac{1}{1+(5x)^2}$.
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State the derivative of $\text{arcsin}(x)$.
State the derivative of $\text{arcsin}(x)$.
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$\frac{1}{\text{sqrt}(1-x^2)}$. Standard derivative formula for inverse sine function.
$\frac{1}{\text{sqrt}(1-x^2)}$. Standard derivative formula for inverse sine function.
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Identify the derivative of $\text{arccos}(3x)$ with respect to $x$.
Identify the derivative of $\text{arccos}(3x)$ with respect to $x$.
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$-\frac{3}{\sqrt{1-9x^2}}$. Apply chain rule: derivative of $3x$ times $-\frac{1}{\sqrt{1-(3x)^2}}$.
$-\frac{3}{\sqrt{1-9x^2}}$. Apply chain rule: derivative of $3x$ times $-\frac{1}{\sqrt{1-(3x)^2}}$.
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State the derivative of $\text{arccsc}(x)$.
State the derivative of $\text{arccsc}(x)$.
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$ -\frac{1}{|x| \sqrt{x^2 - 1}} $ Standard derivative formula for inverse cosecant function.
$ -\frac{1}{|x| \sqrt{x^2 - 1}} $ Standard derivative formula for inverse cosecant function.
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Find the derivative of $y = \text{arccos}(\frac{x}{2})$ at $x = 1$.
Find the derivative of $y = \text{arccos}(\frac{x}{2})$ at $x = 1$.
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$ -\frac{1}{\sqrt{3}} $. Use chain rule with $ \frac{d}{dx}[\frac{x}{2}] = \frac{1}{2} $ and evaluate at $x = 1$.
$ -\frac{1}{\sqrt{3}} $. Use chain rule with $ \frac{d}{dx}[\frac{x}{2}] = \frac{1}{2} $ and evaluate at $x = 1$.
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Determine the derivative of $y = \text{arccos}(\text{cos}(x))$.
Determine the derivative of $y = \text{arccos}(\text{cos}(x))$.
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$-1$. Inverse function cancels on principal domain.
$-1$. Inverse function cancels on principal domain.
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Identify the derivative of $\text{arcsec}(6x)$ with respect to $x$.
Identify the derivative of $\text{arcsec}(6x)$ with respect to $x$.
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$\frac{6}{|6x| \sqrt{36x^2 - 1}}$. Apply chain rule with arcsec derivative formula.
$\frac{6}{|6x| \sqrt{36x^2 - 1}}$. Apply chain rule with arcsec derivative formula.
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Find the derivative of $y = \text{arctan}(x^3)$ at $x = 1$.
Find the derivative of $y = \text{arctan}(x^3)$ at $x = 1$.
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$3$. Use chain rule with $\frac{d}{dx}[x^3] = 3x^2$ and evaluate at $x = 1$.
$3$. Use chain rule with $\frac{d}{dx}[x^3] = 3x^2$ and evaluate at $x = 1$.
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Determine the derivative of $y = \text{arccsc}(\text{csc}(x))$.
Determine the derivative of $y = \text{arccsc}(\text{csc}(x))$.
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$-1$. Inverse function cancels on principal domain.
$-1$. Inverse function cancels on principal domain.
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Find the derivative of $y = \text{arccsc}(x^2)$ at $x = 2$.
Find the derivative of $y = \text{arccsc}(x^2)$ at $x = 2$.
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$-\frac{1}{4\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[x^2] = 2x$ and evaluate at $x = 2$.
$-\frac{1}{4\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[x^2] = 2x$ and evaluate at $x = 2$.
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State the derivative of $\text{arccos}(x)$.
State the derivative of $\text{arccos}(x)$.
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$- rac{1}{\sqrt{1 - x^2}}$. Standard derivative formula for inverse cosine function.
$- rac{1}{\sqrt{1 - x^2}}$. Standard derivative formula for inverse cosine function.
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