Derivatives of Reciprocal Trig Functions - AP Calculus BC
Card 1 of 30
State the formula for the derivative of cotangent.
State the formula for the derivative of cotangent.
Tap to reveal answer
$ - \csc^2(x) $. Memorized derivative formula.
$ - \csc^2(x) $. Memorized derivative formula.
← Didn't Know|Knew It →
Identify the derivative: $\frac{d}{dx} \sec(x)$
Identify the derivative: $\frac{d}{dx} \sec(x)$
Tap to reveal answer
$\sec(x)\tan(x)$. Standard derivative of secant.
$\sec(x)\tan(x)$. Standard derivative of secant.
← Didn't Know|Knew It →
What is the derivative of $\sec(x)$?
What is the derivative of $\sec(x)$?
Tap to reveal answer
$\sec(x)\tan(x)$. Derivative formula for secant function.
$\sec(x)\tan(x)$. Derivative formula for secant function.
← Didn't Know|Knew It →
What is the slope of the tangent line to $y = \cot(x)$ at $x = \frac{\pi}{2}$?
What is the slope of the tangent line to $y = \cot(x)$ at $x = \frac{\pi}{2}$?
Tap to reveal answer
$-1$. Derivative equals slope at given point.
$-1$. Derivative equals slope at given point.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx} \cot(x)$ at $x = \frac{\pi}{4}$.
Evaluate $\frac{d}{dx} \cot(x)$ at $x = \frac{\pi}{4}$.
Tap to reveal answer
$-2$. $-\csc^2(\frac{\pi}{4}) = -(\sqrt{2})^2 = -2$.
$-2$. $-\csc^2(\frac{\pi}{4}) = -(\sqrt{2})^2 = -2$.
← Didn't Know|Knew It →
Identify the derivative: $\frac{d}{dx} \cot(x)$
Identify the derivative: $\frac{d}{dx} \cot(x)$
Tap to reveal answer
$-\csc^2(x)$. Standard derivative of cotangent.
$-\csc^2(x)$. Standard derivative of cotangent.
← Didn't Know|Knew It →
Find the derivative of $y = \sec(x)$.
Find the derivative of $y = \sec(x)$.
Tap to reveal answer
$\frac{dy}{dx} = \sec(x)\tan(x)$. Apply secant derivative rule.
$\frac{dy}{dx} = \sec(x)\tan(x)$. Apply secant derivative rule.
← Didn't Know|Knew It →
Determine the derivative of $f(x) = \cot^2(x)$.
Determine the derivative of $f(x) = \cot^2(x)$.
Tap to reveal answer
$-2\cot(x)\csc^2(x)$. Use chain rule: $2\cot(x) \cdot (-\csc^2(x))$.
$-2\cot(x)\csc^2(x)$. Use chain rule: $2\cot(x) \cdot (-\csc^2(x))$.
← Didn't Know|Knew It →
Find the derivative of $y = \tan(x)$.
Find the derivative of $y = \tan(x)$.
Tap to reveal answer
$\frac{dy}{dx} = \sec^2(x)$. Apply tangent derivative rule.
$\frac{dy}{dx} = \sec^2(x)$. Apply tangent derivative rule.
← Didn't Know|Knew It →
Differentiate: $\csc(x)$
Differentiate: $\csc(x)$
Tap to reveal answer
$ -\csc(x)\cot(x) $. Use derivative formula for $\csc(x)$.
$ -\csc(x)\cot(x) $. Use derivative formula for $\csc(x)$.
← Didn't Know|Knew It →
Determine the derivative of $f(x) = \tan^2(x)$.
Determine the derivative of $f(x) = \tan^2(x)$.
Tap to reveal answer
$2\tan(x)\sec^2(x)$. Use chain rule: $2\tan(x) \cdot \sec^2(x)$.
$2\tan(x)\sec^2(x)$. Use chain rule: $2\tan(x) \cdot \sec^2(x)$.
← Didn't Know|Knew It →
Differentiate: $\tan(x)$
Differentiate: $\tan(x)$
Tap to reveal answer
$\sec^2(x)$. Use derivative formula for $\tan(x)$.
$\sec^2(x)$. Use derivative formula for $\tan(x)$.
← Didn't Know|Knew It →
State the formula for the derivative of tangent.
State the formula for the derivative of tangent.
Tap to reveal answer
$\sec^2(x)$. Memorized derivative formula.
$\sec^2(x)$. Memorized derivative formula.
← Didn't Know|Knew It →
What is the slope of the tangent line to $y = \tan(x)$ at $x = \frac{\pi}{4}$?
What is the slope of the tangent line to $y = \tan(x)$ at $x = \frac{\pi}{4}$?
Tap to reveal answer
$2$. Derivative equals slope at given point.
$2$. Derivative equals slope at given point.
← Didn't Know|Knew It →
Differentiate: $\sec(x)$
Differentiate: $\sec(x)$
Tap to reveal answer
$\sec(x)\tan(x)$. Use derivative formula for $\sec(x)$.
$\sec(x)\tan(x)$. Use derivative formula for $\sec(x)$.
← Didn't Know|Knew It →
State the formula for the derivative of secant.
State the formula for the derivative of secant.
Tap to reveal answer
$\sec(x)\tan(x)$. Memorized derivative formula.
$\sec(x)\tan(x)$. Memorized derivative formula.
← Didn't Know|Knew It →
Compute the derivative of $f(x) = \sec(x)$ at $x = \frac{\pi}{3}$.
Compute the derivative of $f(x) = \sec(x)$ at $x = \frac{\pi}{3}$.
Tap to reveal answer
$2\sqrt{3}$. $\sec(\frac{\pi}{3})\tan(\frac{\pi}{3}) = 2 \cdot \sqrt{3} = 2\sqrt{3}$.
$2\sqrt{3}$. $\sec(\frac{\pi}{3})\tan(\frac{\pi}{3}) = 2 \cdot \sqrt{3} = 2\sqrt{3}$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx} \csc(x)$ at $x = \frac{\pi}{2}$.
Evaluate $\frac{d}{dx} \csc(x)$ at $x = \frac{\pi}{2}$.
Tap to reveal answer
$0$. $-\csc(\frac{\pi}{2})\cot(\frac{\pi}{2}) = -1 \cdot 0 = 0$.
$0$. $-\csc(\frac{\pi}{2})\cot(\frac{\pi}{2}) = -1 \cdot 0 = 0$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx} \sec(x)$ at $x = 0$.
Evaluate $\frac{d}{dx} \sec(x)$ at $x = 0$.
Tap to reveal answer
$0$. $\sec(0)\tan(0) = 1 \cdot 0 = 0$.
$0$. $\sec(0)\tan(0) = 1 \cdot 0 = 0$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx} \tan(x)$ at $x = 0$.
Evaluate $\frac{d}{dx} \tan(x)$ at $x = 0$.
Tap to reveal answer
$1$. $\sec^2(0) = 1^2 = 1$.
$1$. $\sec^2(0) = 1^2 = 1$.
← Didn't Know|Knew It →
Differentiate: $\cot(x)$
Differentiate: $\cot(x)$
Tap to reveal answer
$-\csc^2(x)$. Use derivative formula for $\cot(x)$.
$-\csc^2(x)$. Use derivative formula for $\cot(x)$.
← Didn't Know|Knew It →
Find the derivative of $y = \csc(x)$.
Find the derivative of $y = \csc(x)$.
Tap to reveal answer
$\frac{dy}{dx} = -\csc(x)\cot(x)$. Apply cosecant derivative rule.
$\frac{dy}{dx} = -\csc(x)\cot(x)$. Apply cosecant derivative rule.
← Didn't Know|Knew It →
What is the derivative of $\tan(x)$?
What is the derivative of $\tan(x)$?
Tap to reveal answer
$\sec^2(x)$. Derivative formula for tangent function.
$\sec^2(x)$. Derivative formula for tangent function.
← Didn't Know|Knew It →
What is the derivative of $\cot(x)$?
What is the derivative of $\cot(x)$?
Tap to reveal answer
$-\csc^2(x)$. Derivative formula for cotangent function.
$-\csc^2(x)$. Derivative formula for cotangent function.
← Didn't Know|Knew It →
What is the derivative of $\csc(x)$?
What is the derivative of $\csc(x)$?
Tap to reveal answer
$-\csc(x)\cot(x)$. Derivative formula for cosecant function.
$-\csc(x)\cot(x)$. Derivative formula for cosecant function.
← Didn't Know|Knew It →
Compute the derivative of $f(x) = \tan(x)$ at $x = \frac{\pi}{4}$.
Compute the derivative of $f(x) = \tan(x)$ at $x = \frac{\pi}{4}$.
Tap to reveal answer
$2$. $\sec^2(\frac{\pi}{4}) = (\sqrt{2})^2 = 2$.
$2$. $\sec^2(\frac{\pi}{4}) = (\sqrt{2})^2 = 2$.
← Didn't Know|Knew It →
Find the derivative of $y = \cot(x)$.
Find the derivative of $y = \cot(x)$.
Tap to reveal answer
$\frac{dy}{dx} = -\csc^2(x)$. Apply cotangent derivative rule.
$\frac{dy}{dx} = -\csc^2(x)$. Apply cotangent derivative rule.
← Didn't Know|Knew It →
Identify the derivative: $\frac{d}{dx} \csc(x)$
Identify the derivative: $\frac{d}{dx} \csc(x)$
Tap to reveal answer
$-\csc(x)\cot(x)$. Standard derivative of cosecant.
$-\csc(x)\cot(x)$. Standard derivative of cosecant.
← Didn't Know|Knew It →
Identify the derivative: $\frac{d}{dx} \tan(x)$
Identify the derivative: $\frac{d}{dx} \tan(x)$
Tap to reveal answer
$\sec^2(x)$. Standard derivative of tangent.
$\sec^2(x)$. Standard derivative of tangent.
← Didn't Know|Knew It →
State the formula for the derivative of cosecant.
State the formula for the derivative of cosecant.
Tap to reveal answer
$-\csc(x)\cot(x)$. Memorized derivative formula.
$-\csc(x)\cot(x)$. Memorized derivative formula.
← Didn't Know|Knew It →