Derivatives of inverse trigonometric functions:

• What do inverse trigonometric functions return?

  The angle
• The principal value range of $\arccos x$ is $0 ≤ y ≤ π$


  True
• The range of $\csc^{ - 1} x$ includes $y ≠ 0$


  True
• What is the derivative of $e^{x}$?


  $e^{x}$
• What is the derivative of $\arcsin x$?


  \frac{1}{\sqrt{1 - x^{2}}}</latex>
• Match the inverse trigonometric function with its notation:

  Arcsine ↔️ $\sin^{ - 1}(x)$ or $\arcsin(x)$

  Arccosine ↔️ $\cos^{ - 1}(x)$ or $\arccos(x)$

  Arctangent ↔️ $\tan^{ - 1}(x)$ or $\arctan(x)$
• What is the range of $\arctan x$?


  $- \frac{\pi}{2} < y < \frac{\pi}{2}$
• Match the trigonometric function with its derivative:

  1️⃣ $\sin x$

  2️⃣ $\cos x$

  3️⃣ $\tan x$
• The derivative of $\ln x$ is $\frac{1}{x}$
• Steps to derive the derivative of $\arcsin x$


  1️⃣ Rewrite $y =$$\arcsin x$ as $x =$$\sin y$

  2️⃣ Differentiate both sides with respect to $x$

  3️⃣ Solve for $\frac{dy}{dx}$

  4️⃣ Find $\cos y$ in terms of $x$

  5️⃣ Substitute $\cos y$ into the expression for $\frac{dy}{dx}$
• The range of $\csc^{ - 1} x$ is $- \frac{\pi}{2} ≤ y ≤ \frac{\pi}{2}$, y ≠ 0
• The domain of the arccosine function is -1 ≤ x ≤ 1


  True
• Match each inverse trigonometric function with its range:

  Arcsine ↔️ -π/2 ≤ y ≤ π/2

  Arccosine ↔️ 0 ≤ y ≤ π

  Arctangent ↔️ -π/2 < y < π/2

  Arccotangent ↔️ 0 < y < π

  Arcsecant ↔️ 0 ≤ y ≤ π, y ≠ π/2

  Arccosecant ↔️ -π/2 ≤ y ≤ π/2, y ≠ 0
• The derivative of sin x is cos x
• The derivative of eˣ is eˣ

  True
• What is the derivative of arcsin x?

  \frac{1}{\sqrt{1 - x^{2}}}</latex>
• The domain of $\arcsin x$ is -1 ≤ x ≤ 1
• The domain of \sec^{-1} x</latex> is x ≤ -1 or x ≥ 1
• Match the trigonometric function with its derivative:

  $\sin x$ ↔️ $\cos x$

  $\cos x$ ↔️ $- \sin x$

  $\tan x$ ↔️ $\sec^{2} x$
• Inverse trigonometric functions find angles corresponding to trigonometric values.

  True
• The value of $\cos y$ in the derivative of $\arcsin x$ is $\sqrt{1 - x^{2}}$


  True
• What is the notation for the arcsine function?

  asin(x) or sin⁻¹(x)
• What is the domain of the arctangent function?

  All real numbers
• What is the derivative of xⁿ according to the power rule?

  nxⁿ⁻¹
• Match each function with its derivative:

  xⁿ ↔️ nxⁿ⁻¹

  sin x ↔️ cos x

  cos x ↔️ -sin x

  tan x ↔️ sec² x

  ln x ↔️ 1/x
• Steps to derive the derivative of arccos x using implicit differentiation and the chain rule:

  1️⃣ Rewrite y = arccos x as x = cos y

  2️⃣ Differentiate both sides with respect to x

  3️⃣ Solve for dy/dx

  4️⃣ Find sin y using a right triangle and the Pythagorean theorem

  5️⃣ Substitute sin y into dy/dx
• The derivative of $\arccos x$ is $- \frac{1}{\sqrt{1 - x^{2}}}$
• The derivative of \arccos x</latex> is $- \frac{1}{\sqrt{1 - x^{2}}}$
• What is $\cos y$ equal to in the equation $x =$$\cos y$?


  $\frac{x}{1}$
• The value of $\sin y$ in the equation $x =$$\cos y$ is $\sqrt{1 - x^{2}}$


  True
• The derivative of $\arctan x$ is $\frac{1}{\sec^{2} y}$
• The derivative of $\arctan x$ is $\frac{1}{1 + x^{2}}$
• What is the range of $\arccos x$?


  $0 \leq y \leq \pi$
• The equation $y =$$\arcsin x$ can be rewritten as $x =$$\sin y$.


  True
• The derivative of $\arccos x$ is $- \frac{1}{\sqrt{1 - x^{2}}}$
• The sine of $y$ in the derivative of $\arccos x$ is found using the Pythagorean
• The derivative of $\arctan x$ is found using implicit differentiation.


  True
• Match the inverse trigonometric function with its derivative:

  $\csc^{ - 1} x$ ↔️ $- \frac{1}{|x| \sqrt{x^{2} - 1}}$

  $\sec^{ - 1} x$ ↔️ $\frac{1}{|x| \sqrt{x^{2} - 1}}$

  $\cot^{ - 1} x$ ↔️ $- \frac{1}{1 + x^{2}}$
• Match the inverse trigonometric function with its notation:

  Arcsine ↔️ $\sin^{ - 1}(x)$

  Arccosine ↔️ $\cos^{ - 1}(x)$

  Arctangent ↔️ $\tan^{ - 1}(x)$

  Arccotangent ↔️ $\cot^{ - 1}(x)$
• What is the derivative of $x^{3}$?


  $3x^{2}$