Derivatives of Reciprocal Trig Functions - AP Calculus AB
Card 1 of 30
State the derivative of $f(x) = \cot(3x^3)$.
State the derivative of $f(x) = \cot(3x^3)$.
Tap to reveal answer
$-9x^2\csc^2(3x^3)$. Chain rule: multiply by derivative of inner function $9x^2$.
$-9x^2\csc^2(3x^3)$. Chain rule: multiply by derivative of inner function $9x^2$.
← Didn't Know|Knew It →
State the derivative of $\cot(x)$.
State the derivative of $\cot(x)$.
Tap to reveal answer
$ -\csc^2(x) $. Standard derivative formula for cotangent function.
$ -\csc^2(x) $. Standard derivative formula for cotangent function.
← Didn't Know|Knew It →
State the derivative of $g(x) = \sec(x^3)$.
State the derivative of $g(x) = \sec(x^3)$.
Tap to reveal answer
$3x^2\sec(x^3)\tan(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.
$3x^2\sec(x^3)\tan(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.
← Didn't Know|Knew It →
Compute $d/dx$ for $g(x) = \sec(x^2 + 1)$.
Compute $d/dx$ for $g(x) = \sec(x^2 + 1)$.
Tap to reveal answer
$2x\sec(x^2 + 1)\tan(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.
$2x\sec(x^2 + 1)\tan(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.
← Didn't Know|Knew It →
What is the derivative of $f(x) = \cot(3x)$?
What is the derivative of $f(x) = \cot(3x)$?
Tap to reveal answer
$-3\csc^2(3x)$. Chain rule: multiply by derivative of inner function $3$.
$-3\csc^2(3x)$. Chain rule: multiply by derivative of inner function $3$.
← Didn't Know|Knew It →
What is the derivative of $\sec(x)$?
What is the derivative of $\sec(x)$?
Tap to reveal answer
$\sec(x)\tan(x)$. Standard derivative formula for secant function.
$\sec(x)\tan(x)$. Standard derivative formula for secant function.
← Didn't Know|Knew It →
State $d/dx$ for $g(x) = \cot(\sqrt{x})$.
State $d/dx$ for $g(x) = \cot(\sqrt{x})$.
Tap to reveal answer
$-\frac{1}{2\sqrt{x}}\csc^2(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.
$-\frac{1}{2\sqrt{x}}\csc^2(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.
← Didn't Know|Knew It →
Find $d/dx$ for $g(x) = \cot(\ln(x))$.
Find $d/dx$ for $g(x) = \cot(\ln(x))$.
Tap to reveal answer
$-\frac{1}{x}\csc^2(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.
$-\frac{1}{x}\csc^2(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.
← Didn't Know|Knew It →
Compute the derivative of $g(x) = \csc(x^4)$.
Compute the derivative of $g(x) = \csc(x^4)$.
Tap to reveal answer
$-4x^3\csc(x^4)\cot(x^4)$. Chain rule: multiply by derivative of inner function $4x^3$.
$-4x^3\csc(x^4)\cot(x^4)$. Chain rule: multiply by derivative of inner function $4x^3$.
← Didn't Know|Knew It →
Determine the derivative of $y = \tan(3x^2)$.
Determine the derivative of $y = \tan(3x^2)$.
Tap to reveal answer
$6x\sec^2(3x^2)$. Chain rule: multiply by derivative of inner function $6x$.
$6x\sec^2(3x^2)$. Chain rule: multiply by derivative of inner function $6x$.
← Didn't Know|Knew It →
What is the derivative of $f(x) = \tan(x^2 + 1)$?
What is the derivative of $f(x) = \tan(x^2 + 1)$?
Tap to reveal answer
$2x\sec^2(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.
$2x\sec^2(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.
← Didn't Know|Knew It →
Identify the derivative of $y = \tan(x + \pi)$.
Identify the derivative of $y = \tan(x + \pi)$.
Tap to reveal answer
$\sec^2(x + \pi)$. Chain rule: multiply by derivative of inner function $1$.
$\sec^2(x + \pi)$. Chain rule: multiply by derivative of inner function $1$.
← Didn't Know|Knew It →
Determine $d/dx$ for $y = \csc(e^x)$.
Determine $d/dx$ for $y = \csc(e^x)$.
Tap to reveal answer
$-e^x\csc(e^x)\cot(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.
$-e^x\csc(e^x)\cot(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.
← Didn't Know|Knew It →
Find $d/dx$ for $h(x) = \sec(3x + 1)$.
Find $d/dx$ for $h(x) = \sec(3x + 1)$.
Tap to reveal answer
$3\sec(3x + 1)\tan(3x + 1)$. Chain rule: multiply by derivative of inner function $3$.
$3\sec(3x + 1)\tan(3x + 1)$. Chain rule: multiply by derivative of inner function $3$.
← Didn't Know|Knew It →
Calculate the derivative of $y = \csc(4x)$.
Calculate the derivative of $y = \csc(4x)$.
Tap to reveal answer
$-4\csc(4x)\cot(4x)$. Chain rule: multiply by derivative of inner function $4$.
$-4\csc(4x)\cot(4x)$. Chain rule: multiply by derivative of inner function $4$.
← Didn't Know|Knew It →
State the derivative of $g(x) = \cot(2x^3)$.
State the derivative of $g(x) = \cot(2x^3)$.
Tap to reveal answer
$-6x^2\csc^2(2x^3)$. Chain rule: multiply by derivative of inner function $6x^2$.
$-6x^2\csc^2(2x^3)$. Chain rule: multiply by derivative of inner function $6x^2$.
← Didn't Know|Knew It →
What is the derivative of $f(x) = \tan(5x^2)$?
What is the derivative of $f(x) = \tan(5x^2)$?
Tap to reveal answer
$10x\sec^2(5x^2)$. Chain rule: multiply by derivative of inner function $10x$.
$10x\sec^2(5x^2)$. Chain rule: multiply by derivative of inner function $10x$.
← Didn't Know|Knew It →
State the derivative of $f(x) = \cot(3x^3)$.
State the derivative of $f(x) = \cot(3x^3)$.
Tap to reveal answer
$-9x^2\csc^2(3x^3)$. Chain rule: multiply by derivative of inner function $9x^2$.
$-9x^2\csc^2(3x^3)$. Chain rule: multiply by derivative of inner function $9x^2$.
← Didn't Know|Knew It →
Find the derivative of $h(x) = \csc(\ln(x))$.
Find the derivative of $h(x) = \csc(\ln(x))$.
Tap to reveal answer
$-\frac{1}{x}\csc(\ln(x))\cot(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.
$-\frac{1}{x}\csc(\ln(x))\cot(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.
← Didn't Know|Knew It →
Calculate $d/dx$ for $g(x) = \sec(x^5)$.
Calculate $d/dx$ for $g(x) = \sec(x^5)$.
Tap to reveal answer
$5x^4\sec(x^5)\tan(x^5)$. Chain rule: multiply by derivative of inner function $5x^4$.
$5x^4\sec(x^5)\tan(x^5)$. Chain rule: multiply by derivative of inner function $5x^4$.
← Didn't Know|Knew It →
What is the derivative of $y = \cot(e^x)$?
What is the derivative of $y = \cot(e^x)$?
Tap to reveal answer
$-e^x\csc^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.
$-e^x\csc^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.
← Didn't Know|Knew It →
Find the derivative of $f(x) = \tan(e^x)$.
Find the derivative of $f(x) = \tan(e^x)$.
Tap to reveal answer
$e^x\sec^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.
$e^x\sec^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.
← Didn't Know|Knew It →
Determine the derivative of $y = \csc(x^2)$.
Determine the derivative of $y = \csc(x^2)$.
Tap to reveal answer
$-2x\csc(x^2)\cot(x^2)$. Chain rule: multiply by derivative of inner function $2x$.
$-2x\csc(x^2)\cot(x^2)$. Chain rule: multiply by derivative of inner function $2x$.
← Didn't Know|Knew It →
Compute the derivative of $h(x) = \sec(\ln(x))$.
Compute the derivative of $h(x) = \sec(\ln(x))$.
Tap to reveal answer
$\frac{1}{x}\sec(\ln(x))\tan(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.
$\frac{1}{x}\sec(\ln(x))\tan(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.
← Didn't Know|Knew It →
Determine the derivative of $y = \csc(4x^2)$.
Determine the derivative of $y = \csc(4x^2)$.
Tap to reveal answer
$-8x\csc(4x^2)\cot(4x^2)$. Chain rule: multiply by derivative of inner function $8x$.
$-8x\csc(4x^2)\cot(4x^2)$. Chain rule: multiply by derivative of inner function $8x$.
← Didn't Know|Knew It →
Find $d/dx$ for $f(x) = \sec(\sin(x))$.
Find $d/dx$ for $f(x) = \sec(\sin(x))$.
Tap to reveal answer
$\cos(x)\sec(\sin(x))\tan(\sin(x))$. Chain rule: multiply by derivative of $\sin(x)$ which is $\cos(x)$.
$\cos(x)\sec(\sin(x))\tan(\sin(x))$. Chain rule: multiply by derivative of $\sin(x)$ which is $\cos(x)$.
← Didn't Know|Knew It →
Compute the derivative of $g(x) = \sec(5x)$.
Compute the derivative of $g(x) = \sec(5x)$.
Tap to reveal answer
$5\sec(5x)\tan(5x)$. Chain rule: multiply by derivative of inner function $5$.
$5\sec(5x)\tan(5x)$. Chain rule: multiply by derivative of inner function $5$.
← Didn't Know|Knew It →
Find $d/dx$ for $f(x) = \sec^2(x)$.
Find $d/dx$ for $f(x) = \sec^2(x)$.
Tap to reveal answer
$2\sec(x)\sec(x)\tan(x)$. Chain rule applied to $\sec^2(x) = 2\sec(x) \cdot \sec(x)\tan(x)$.
$2\sec(x)\sec(x)\tan(x)$. Chain rule applied to $\sec^2(x) = 2\sec(x) \cdot \sec(x)\tan(x)$.
← Didn't Know|Knew It →
What is the derivative of $y = \cot(x^3)$?
What is the derivative of $y = \cot(x^3)$?
Tap to reveal answer
$-3x^2\csc^2(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.
$-3x^2\csc^2(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.
← Didn't Know|Knew It →
Determine $d/dx$ for $h(x) = \csc(2x)$.
Determine $d/dx$ for $h(x) = \csc(2x)$.
Tap to reveal answer
$-2\csc(2x)\cot(2x)$. Chain rule: multiply by derivative of inner function $2$.
$-2\csc(2x)\cot(2x)$. Chain rule: multiply by derivative of inner function $2$.
← Didn't Know|Knew It →