Inverse trigonometric functions undo the basic trig functions on restricted domains, so they are used whenever an angle must be found from a ratio. Their derivatives are important in calculus because they appear in integration, related rates, curve analysis, and many modeling problems. These formulas also show how algebraic expressions like 1/sqrt(1 - x^2) and 1/(1 + x^2) connect to geometry on a circle or triangle.
Key Facts
- d/dx[arcsin x] = 1/sqrt(1 - x^2), for -1 < x < 1
- d/dx[arccos x] = -1/sqrt(1 - x^2), for -1 < x < 1
- d/dx[arctan x] = 1/(1 + x^2), for all real x
- d/dx[arccot x] = -1/(1 + x^2), for all real x under the common calculus convention
- d/dx[arcsec x] = 1/(|x|sqrt(x^2 - 1)), for |x| > 1
- d/dx[arccsc x] = -1/(|x|sqrt(x^2 - 1)), for |x| > 1
Vocabulary
- Inverse trigonometric function
- A function that returns an angle whose trigonometric value equals the input, using a restricted range so the inverse is a function.
- Domain
- The set of input values for which a function or derivative is defined.
- Range restriction
- A chosen interval of angles that makes a trigonometric function one-to-one so an inverse function can be defined.
- Chain rule
- The derivative rule d/dx[f(g(x))] = f'(g(x))g'(x), used when an inverse trig function contains an inner expression.
- Absolute value
- The distance of a number from zero, which appears in arcsec and arccsc derivatives to handle both positive and negative inputs correctly.
Common Mistakes to Avoid
- Forgetting the chain rule, which gives an incomplete derivative when the input is not just x. For example, d/dx[arctan(3x)] is 3/(1 + 9x^2), not 1/(1 + 9x^2).
- Dropping the negative sign for arccos x or arccot x, which changes the direction of change. Since arccos x decreases as x increases, its derivative must be negative on its domain.
- Ignoring domain restrictions, which can make a formula appear valid where the function or derivative is not defined. For example, d/dx[arcsin x] is not defined at x = 1 because sqrt(1 - x^2) becomes zero.
- Omitting the absolute value in the arcsec and arccsc derivatives, which gives the wrong sign for negative x values. The correct denominator is |x|sqrt(x^2 - 1), not x sqrt(x^2 - 1).
Practice Questions
- 1 Find d/dx[arcsin(4x)]. State the interval of x-values where the derivative formula is valid.
- 2 Find d/dx[arcsec(2x)] and simplify the result. State the values of x where the derivative is defined.
- 3 Explain why the derivatives of arcsin x and arccos x have the same denominator but opposite signs.