Derivatives of Inverse Trigonometric Functions

The inverse trigonometric functions — arcsin, arccos, and arctan — have characteristic derivative formulas. These are important in integration and solving equations.

📜 Core Formulas

$$ \frac{d}{dx}\big(\arcsin x\big) = \frac{1}{\sqrt{1-x^2}} \quad (|x| < 1) $$ $$ \frac{d}{dx}\big(\arccos x\big) = -\frac{1}{\sqrt{1-x^2}} \quad (|x| < 1) $$ $$ \frac{d}{dx}\big(\arctan x\big) = \frac{1}{1+x^2} \quad (x \in \mathbb{R}) $$

💡 Domain restrictions: – $\arcsin(x)$ and $\arccos(x)$ are only defined for $|x|\le 1$. – $\arctan(x)$ is defined for all real $x$. These restrictions carry over to their derivatives.

🪜 Application Strategy

Recognize the inverse trig function: arcsin, arccos, or arctan.
Apply the core formula: Use the derivatives above.
Use the chain rule if the argument is more than just $x$: For example: $\dfrac{d}{dx}\arcsin(u(x)) = \dfrac{u'(x)}{\sqrt{1-(u(x))^2}}$.

✏️ Examples

Example 1: arcsin

$f(x) = \arcsin(x)$

Derivative: $$ f'(x) = \frac{1}{\sqrt{1-x^2}}. $$

Example 2: arccos

$f(x) = \arccos(x)$

Derivative: $$ f'(x) = -\frac{1}{\sqrt{1-x^2}}. $$

Example 3: arctan

$f(x) = \arctan(x)$

Derivative: $$ f'(x) = \frac{1}{1+x^2}. $$

Example 4: Chain Rule with arcsin

$f(x) = \arcsin(2x)$ with $|2x|\le 1$

Outer derivative: $\dfrac{1}{\sqrt{1-(2x)^2}}$
Inner derivative: $2$

Result: $$ f'(x) = \frac{2}{\sqrt{1-4x^2}}. $$

Example 5: Chain Rule with arccos

$f(x) = \arccos(x^2)$ with $|x^2|\le 1$

Outer derivative: $-\dfrac{1}{\sqrt{1-(x^2)^2}}$
Inner derivative: $2x$

Result: $$ f'(x) = -\frac{2x}{\sqrt{1-x^4}}. $$

Example 6: Chain Rule with arctan

$f(x) = \arctan(3x-1)$

Outer derivative: $\dfrac{1}{1+(3x-1)^2}$
Inner derivative: $3$

Result: $$ f'(x) = \frac{3}{1+(3x-1)^2}. $$