FAC1004 Tutorial 7 — Derivatives of Inverse Trig Functions

Practice problems on differentiation of inverse trigonometric functions.

Topics Covered

  • Derivative formulas for inverse trig functions
  • Chain rule applications
  • Product and quotient rules
  • Implicit differentiation
  • Introduction to hyperbolic identities

Problem Set

  1. Solve Inverse Trig Equation: Solve and verify $$\sin^{-1}(2x) + \frac{\pi}{4} = \tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$$

  2. Prove Derivative Formulas: Show that:

    • $[\cos^{-1} x]' = -\frac{1}{\sqrt{1-x^2}}$
    • $[\tan^{-1} x]' = \frac{1}{1+x^2}$
    • $[\csc^{-1} x]' = -\frac{1}{|x|\sqrt{x^2-1}}$

  3. Basic Derivatives: Find first derivatives of:

    • $y = \sin^{-1}(3x)$
    • $y = \cos^{-1}\left(\frac{x+1}{2}\right)$
    • $y = \sin^{-1}\left(\frac{1}{x}\right)$
    • $y = \sec^{-1}(x^2)$
    • $y = \tan^{-1}(e^{2x})$
    • $y = \ln(x^2)\sec^{-1}(4x^3)$

  4. Advanced Differentiation: Differentiate:

    • $y = x + \sin^{-1}(e^{-x})$
    • $y = \tan^{-1}(x^2)\csc^{-1}(\ln x)$
    • $y = \frac{\cos^{-1}(2x)}{3x - e^x}$
    • $y = \frac{e^{x^2}\sin^{-1}(5x)}{\ln(x^2)\tan x}$

  5. Hyperbolic Identity: Show that $\cosh x + \sinh x = e^x$

  6. Hyperbolic Identities: Prove $\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ and $\text{sech } x = \frac{2}{e^x + e^{-x}}$, then verify $\text{sech}^2 x = 1 - \tanh^2 x$

Related

Source File

TUTORIALS_SET_2526/FAC1004 Tutorial 7 25-26.pdf