3.4 Differentiating Inverse Trigonometric Functions

• The range of arcsin(x) is -π/2 ≤ y ≤ π/2
• Match the inverse trigonometric function with its range:

  arcsin(x) ↔️ -π/2 ≤ y ≤ π/2

  arccos(x) ↔️ 0 ≤ y ≤ π

  arctan(x) ↔️ -π/2 < y < π/2
• The derivative of arccos(x) is $- 1 / √(1 - x^{2})$.


  True
• The derivative formulas for inverse trigonometric functions are derived using the properties of inverse functions and the chain rule.
• The chain rule is needed when differentiating inverse trigonometric functions with composite arguments.


  True
• What are inverse trigonometric functions used to find in a right triangle?

  The angle
• What is the derivative of arcsin(u)?

  $u' / √(1 - u^{2})$
• Inverse trigonometric functions are the inverse functions of sine, cosine, and tangent
• What rule must be applied when differentiating a composite function of x?

  Chain rule
• When differentiating inverse trigonometric functions, what must be applied if 'u' is a function of 'x'?

  Chain rule
• The chain rule is essential for differentiating inverse trigonometric functions when 'u' is a function of 'x'.


  True
• The domain of arcsin(x)</latex> is -1 ≤ x ≤ 1
• The derivative of $arcsin(u)$ is u' / √(1 - u^{2})
• The derivative of $arctan(3x)$ is 3 / (1 + 9x^{2})</latex>.


  True
• The chain rule must be applied when differentiating inverse trigonometric functions where the argument is a function of x.

  True
• The derivative of `g(x) = arccos(x^{2})` is $- 2x / √(1 - x^{4})$
• What is the derivative of $arcsin(u)$?


  u' / √(1 - u^{2})</latex>
• If tan(y) = x, then arctan(x) = y.


  True
• What is the derivative of arcsin(x)?

  $1 / √(1 - x^{2})$
• What is the derivative of arctan(x)?

  $1 / (1 + x^{2})$
• The derivative of arcsin(x) is $1 / √(1 - x^{2})$.


  True
• The derivative of arctan(3x) is 3 / (1 + 9x^{2}).
• The derivative of arctan(3x) is 3 / (1 + 9x^{2})</latex>.


  True
• The derivative of $arcsin(u)$ is u' / √(1 - u^{2})
• The derivative of $arccos(u)$ is - u' / √(1 - u^{2})
• The derivative of `g(x) = arccos(1 / x)` is $1 / (x√(x^{2} - 1))$
• What is the derivative of $arctan(3x)$?


  $3 / (1 + 9x^{2})$
• Match the inverse trigonometric function with its definition:

  arcsin(x) ↔️ Inverse of sine

  arccos(x) ↔️ Inverse of cosine

  arctan(x) ↔️ Inverse of tangent
• If $sin(y) =$$x$, then $arcsin(x) =$$y$.


  True
• What is the derivative of $arctan(3x)$?


  $3 / (1 + 9x^{2})$
• To differentiate inverse trigonometric functions, we can use specific formulas
• The derivative of `f(x) = arcsin(5x)` is $5 / √(1 - 25x^{2})$
• Steps to differentiate an expression involving inverse trigonometric functions

  1️⃣ Identify the inverse trigonometric function

  2️⃣ Apply the appropriate differentiation formula

  3️⃣ If the argument is a function of x, apply the chain rule

  4️⃣ Simplify the result
• The derivative of `f(x) = arcsin(3x^{2})` is $6x / √(1 - 9x^{4})$