Overview
This lesson covers how to differentiate inverse trigonometric functions using standard formulas, with step-by-step examples and simplification strategies.
Derivatives of Inverse Trigonometric Functions
- The derivative of $\arcsin(u)$ is $u' / \sqrt{1 - u^2}$.
- The derivative of $\arccos(u)$ is $-u' / \sqrt{1 - u^2}$.
- The derivative of $\arctan(u)$ is $u' / (1 + u^2)$.
- The derivative of $\arcsec(u)$ is $u' / [|u| \sqrt{u^2 - 1}]$.
Example Problems & Solutions
Example 1: Derivative of $\arcsin(x^3)$
- Let $u = x^3$; thus, $u' = 3x^2$.
- Derivative: $3x^2 / \sqrt{1 - x^6}$.
Example 2: Derivative of $\arccos(5x - 9)$
- Let $u = 5x - 9$; thus, $u' = 5$.
- Derivative: $-5 / \sqrt{1 - (5x - 9)^2}$.
Example 3: Derivative of $\arctan(\sqrt{x})$
- Let $u = \sqrt{x} = x^{1/2}$; thus, $u' = 1/(2\sqrt{x})$.
- Derivative: $1 / [2\sqrt{x}(1 + x)]$.
- Alternatively, write as $\sqrt{x} / [2x(1 + x)]$ after rationalizing.
Example 4: Derivative of $\arcsec(x^4)$
- Let $u = x^4$; thus, $u' = 4x^3$.
- Derivative: $4x^3 / [|x^4| \sqrt{x^8 - 1}]$.
- Simplifies to $4 / [x \sqrt{x^8 - 1}]$.
Key Terms & Definitions
- Inverse Trigonometric Function — A function that reverses a trigonometric function, such as arcsin, arccos, arctan.
- u prime ($u'$) — The derivative of the inner function $u$ with respect to $x$.
- Absolute Value ($|u|$) — Represents the non-negative value of $u$.
Action Items / Next Steps
- Review and memorize the four main derivative formulas for inverse trig functions.
- Practice differentiating more composite inverse trig functions.
- Check textbook for additional derivative formulas and exercises.