Derivative of Inverse Trigonometric Functions

Derivative of Inverse Trigonometric Functions

Key Derivatives

• Arcsine Derivative: $$\frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1 - x^2}}$$

• Arccosine Derivative: $$\frac{d}{dx}[\cos^{-1} x] = -\frac{1}{\sqrt{1 - x^2}}$$

• Arctangent Derivative: $$\frac{d}{dx}[\tan^{-1} x] = \frac{1}{1 + x^2}$$

• Arccosecant Derivative: $$\frac{d}{dx}[\csc^{-1} x] = -\frac{1}{|x|\sqrt{x^2 - 1}}$$

Differentiation Examples

• Chain Rule Applications:

  • $y = \sin^{-1}(3x)$
    $$y' = \frac{1}{\sqrt{1 - (3x)^2}} \cdot 3 = \frac{3}{\sqrt{1 - 9x^2}}$$
  • $y = \tan^{-1}(e^{2x})$
    $$y' = \frac{1}{1 + (e^{2x})^2} \cdot 2e^{2x} = \frac{2e^{2x}}{1 + e^{4x}}$$

• Product Rule:

  • $y = x + \sin^{-1}(e^{2x})$
    $$y' = 1 + \frac{1}{\sqrt{1 - e^{4x}}} \cdot 2e^{2x} = 1 + \frac{2e^{2x}}{\sqrt{1 - e^{4x}}}$$

Tutorial Tips

• Always check the domain of the inverse trig function before differentiating.
• Use the chain rule when the argument is not simply $x$.
• For inverse secant/cosecant, remember the absolute value in the denominator.

Additional Identities (for verification)

• Hyperbolic identity: $\cosh x + \sinh x = e^x$
• $\tanh x = \frac{e^{2x} - 1}{e^{2x} + 1}$, $\sech x = \frac{2}{e^{2x} + 1}$
• $\sech^2 x = 1 - \tanh^2 x$