Differentiation of Inverse Trig functions

  1. Conceptual Overview
    Standard trigonometric functions (like sine and cosine) are periodic, meaning they repeat their values. Because of this, they are not "one-to-one" and do not have true inverses. To create inverse functions (arcsin, arccos, etc.), mathematicians restrict the domain of the original trig functions so that each output has only one input.

  2. The Derivative Formulas
    If "u" is a function of x, we use the chain rule (multiplying by u' or the derivative of the inside).

  • Arcsine and Arccosine:
    d/dx [arcsin u] = u' / sqrt(1 - u^2)
    d/dx [arccos u] = -u' / sqrt(1 - u^2)

  • Arctangent and Arccotangent:
    d/dx [arctan u] = u' / (1 + u^2)
    d/dx [arccot u] = -u' / (1 + u^2)

  • Arcsecant and Arccosecant:
    d/dx [arcsec u] = u' / (|u| sqrt(u^2 - 1))
    d/dx [arccsc u] = -u' / (|u| sqrt(u^2 - 1))

  1. Key Patterns to Remember

  • The "C" Rule: The derivatives for all inverse trig functions starting with the letter "C" (arccos, arccot, arccsc) are negative.

  • Square Roots: Only the "tan" and "cot" formulas do NOT have a square root in the denominator.

  • Absolute Value: The "sec" and "csc" formulas require an absolute value around the "u" outside the square root.