Differentiating Inverse Trigonometric Functions

Overview

• Focus on differentiating inverse trigonometric functions.
• Key formulas for derivatives of arc sine, arc cosine, arc tangent, and arc secant.

Arc Sine Function

Problem: Derivative of arc sine of (x^3)

• Formula: Derivative of (\text{arc sin}(u)) is (\frac{u'}{\sqrt{1-u^2}}).
• Solution Steps:
  • Identify (u = x^3).
  • Calculate (u' = 3x^2).
  • Plug into formula: [ \text{arc sin}(x^3)' = \frac{3x^2}{\sqrt{1-(x^3)^2}} = \frac{3x^2}{\sqrt{1-x^6}} ]

Arc Cosine Function

Problem: Derivative of arc cosine of (5x - 9)

• Formula: Derivative of (\text{arc cos}(u)) is (-\frac{u'}{\sqrt{1-u^2}}).
• Solution Steps:
  • Identify (u = 5x - 9).
  • Calculate (u' = 5).
  • Plug into formula: [ \text{arc cos}(5x - 9)' = -\frac{5}{\sqrt{1-(5x-9)^2}} ]

Arc Tangent Function

Problem: Derivative of arc tangent of (\sqrt{x})

• Formula: Derivative of (\text{arc tan}(u)) is (\frac{u'}{1+u^2}).
• Solution Steps:
  • Identify (u = \sqrt{x} = x^{1/2}).
  • Calculate (u' = \frac{1}{2\sqrt{x}}).
  • Plug into formula and simplify:
    • [ \text{arc tan}(\sqrt{x})' = \frac{1}{2\sqrt{x}(1+x)} ]
    • Simplified to: [ \frac{\sqrt{x}}{2x(1+x)} ]

Arc Secant Function

Problem: Derivative of arc secant of (x^4)

• Formula: Derivative of (\text{arc sec}(u)) is (\frac{u'}{|u|\sqrt{u^2-1}}).
• Solution Steps:
  • Identify (u = x^4).
  • Calculate (u' = 4x^3).
  • Plug into formula and simplify:
    • [ \text{arc sec}(x^4)' = \frac{4x^3}{|x^4|\sqrt{x^8-1}} ]
    • Simplified to: [ \frac{4}{x\sqrt{x^8-1}} ]

Summary

• Mastery of formulas is key to differentiating inverse trigonometric functions.
• Each function's derivative requires identifying (u) and (u'), then applying the specific formula.
• Simplification of expressions is needed for final solutions.