Derivatives Involving Arcsin & Arccos

Derivative Formulas for Inverse Trig

  • Basic (single variable):
    • (ddxarcsin⁡x=11−x2)(\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}})
    • (ddxarccos⁡x=−11−x2)(\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}})
  • Composite (apply Chain Rule, u=u(x)u=u(x)):
    • (ddxarcsin⁡u=u′1−u2)(\frac{d}{dx}\arcsin u = \frac{u'}{\sqrt{1-u^2}})
    • (ddxarccos⁡u=−u′1−u2)(\frac{d}{dx}\arccos u = -\frac{u'}{\sqrt{1-u^2}})

Chain-Rule Awareness

  • Always check if the inverse-trig expression has an inner function u(x)u(x).
  • If not composite (i.e., u=xu=x), chain rule still works because u′=1u'=1.

Example 1: f(x)=3arcsin⁡xf(x)=3\arcsin x

  • Not composite (inner function xx).
  • Derivative:
    • f′(x)=3⋅11−x2=31−x2f'(x)=3\cdot\frac{1}{\sqrt{1-x^2}}=\frac{3}{\sqrt{1-x^2}}

Example 2: g(x)=4arccos⁡(3x2)g(x)=4\arccos(3x^2)

  • Composite with inner function u=3x2u=3x^2 (so u′=6xu'=6x).
  • Apply chain rule:
    • g′(x)=4(−11−u2)u′g'(x)=4\left(-\frac{1}{\sqrt{1-u^2}}\right)u'
    • Substitute u,u′u, u': g′(x)=4(−11−(3x2)2)(6x)g'(x)=4\left(-\frac{1}{\sqrt{1-(3x^2)^2}}\right)(6x)
    • Simplify: g′(x)=−24x1−9x4g'(x)=\frac{-24x}{\sqrt{1-9x^4}}

Key Takeaways

  • Use composite formulas when an inner function is present.
  • If no inner function, basic formulas suffice (or chain rule with u′=1u'=1).