1. **Problem Statement:** Find the derivatives of all trigonometric functions including simple, inverse, hyperbolic, and inverse hyperbolic functions. 2. **Formulas and Rules:** The derivative of a function $f(x)$ is denoted as $f'(x)$ or $\frac{d}{dx}f(x)$. We use standard differentiation rules and known derivatives of trigonometric functions. 3. **Simple Trigonometric Functions:** - $\frac{d}{dx}\sin x = \cos x$ - $\frac{d}{dx}\cos x = -\sin x$ - $\frac{d}{dx}\tan x = \sec^2 x$ - $\frac{d}{dx}\cot x = -\csc^2 x$ - $\frac{d}{dx}\sec x = \sec x \tan x$ - $\frac{d}{dx}\csc x = -\csc x \cot x$ 4. **Inverse Trigonometric Functions:** - $\frac{d}{dx}\sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}$ - $\frac{d}{dx}\cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}}$ - $\frac{d}{dx}\tan^{-1} x = \frac{1}{1 + x^2}$ - $\frac{d}{dx}\cot^{-1} x = -\frac{1}{1 + x^2}$ - $\frac{d}{dx}\sec^{-1} x = \frac{1}{|x|\sqrt{x^2 - 1}}$ - $\frac{d}{dx}\csc^{-1} x = -\frac{1}{|x|\sqrt{x^2 - 1}}$ 5. **Hyperbolic Functions:** - $\frac{d}{dx}\sinh x = \cosh x$ - $\frac{d}{dx}\cosh x = \sinh x$ - $\frac{d}{dx}\tanh x = \operatorname{sech}^2 x$ - $\frac{d}{dx}\coth x = -\operatorname{csch}^2 x$ - $\frac{d}{dx}\operatorname{sech} x = -\operatorname{sech} x \tanh x$ - $\frac{d}{dx}\operatorname{csch} x = -\operatorname{csch} x \coth x$ 6. **Inverse Hyperbolic Functions:** - $\frac{d}{dx}\sinh^{-1} x = \frac{1}{\sqrt{x^2 + 1}}$ - $\frac{d}{dx}\cosh^{-1} x = \frac{1}{\sqrt{x^2 - 1}}$ for $x > 1$ - $\frac{d}{dx}\tanh^{-1} x = \frac{1}{1 - x^2}$ for $|x| < 1$ - $\frac{d}{dx}\coth^{-1} x = \frac{1}{1 - x^2}$ for $|x| > 1$ - $\frac{d}{dx}\operatorname{sech}^{-1} x = -\frac{1}{x \sqrt{1 - x^2}}$ for $0 < x < 1$ - $\frac{d}{dx}\operatorname{csch}^{-1} x = -\frac{1}{|x| \sqrt{1 + x^2}}$ These derivatives are fundamental in calculus and are used extensively in solving problems involving rates of change and integrals involving trigonometric functions.