# Derivatives of Inverse Trigonometric Functions

### Derivatives of Inverse Trigonometric Functions:

1. Inverse Sine Function ($\mathrm{arcsin}\left(x\right)$:

• $\frac{d}{dx}\left[\mathrm{arcsin}\left(x\right)\right]=\frac{1}{\sqrt{1-{x}^{2}}}$for $-1
• Derivative of arcsin(x) is positive for $x$ in the open interval (-1, 1).
2. Inverse Cosine Function ($\mathrm{arccos}\left(x\right)$:

• $\frac{d}{dx}\left[\mathrm{arccos}\left(x\right)\right]=-\frac{1}{\sqrt{1-{x}^{2}}}$ for $-1

Derivative of arccos(x) is negative for $x$ in the open interval (-1, 1).

3. Inverse Tangent Function ($\mathrm{arctan}\left(x\right)$:

• $\frac{d}{dx}\left[\mathrm{arctan}\left(x\right)\right]=\frac{1}{1+{x}^{2}}$
• The derivative of arctan(x) is always positive.

### Derivatives of Other Inverse Trigonometric Functions:

• Derivative of Inverse Cotangent Function ($\text{arccot}\left(x\right)$:

• $\frac{d}{dx}\left[\text{arccot}\left(x\right)\right]=-\frac{1}{1+{x}^{2}}$
• Derivative of Inverse Secant Function ($\text{arcsec}\left(x\right)$:

• $\frac{d}{dx}\left[\text{arcsec}\left(x\right)\right]=\frac{1}{\mathrm{\mid }x\mathrm{\mid }\cdot \sqrt{{x}^{2}-1}}$ for $\mathrm{\mid }x\mathrm{\mid }>1$
•
• Derivative of Inverse Cosecant Function ($\text{arccsc}\left(x\right)$:

• $\frac{d}{dx}\left[\text{arccsc}\left(x\right)\right]=-\frac{1}{\mathrm{\mid }x\mathrm{\mid }\cdot \sqrt{{x}^{2}-1}}$
• for $\mathrm{\mid }x\mathrm{\mid }>1$

### Chain Rule with Inverse Trigonometric Functions:

Chain Rule for Inverse Trigonometric Functions:

• When the inverse trigonometric function is nested within another function, the chain rule applies.
• Example: $f\left(x\right)=\mathrm{arcsin}\left(2{x}^{2}\right)$
• ${f}^{\mathrm{\prime }}\left(x\right)=\frac{4x}{\sqrt{1-4{x}^{4}}}$
•

### Applications of Inverse Trigonometric Functions in Differentiation:

• Used in integration to solve integrals involving algebraic expressions and square roots.
• Often utilized in physics, engineering, and mathematical modeling to describe inverse relationships in various phenomena.