Calculus
Derivative formulae
| Trigonometry function | Hyperbolic function |
|---|---|
| $\frac{d}{dx} \sin(x) = \cos(x)$ | $\frac{d}{dx} \sinh(x) = \cosh(x)$ |
| $\frac{d}{dx} \cos(x) = - \sin(x)$ | $\frac{d}{dx} \cosh(x) = \sinh(x)$ |
| $\frac{d}{dx} \tan(x) = \sec^{2}(x)$ | $\frac{d}{dx} \tanh(x) = \text{sech}^{2}(x)$ |
| $\frac{d}{dx} \cot(x) = -\text{cosec}^{2}(x)$ | $\frac{d}{dx} \coth(x) = -\text{cosech}^{2}(x)$ |
| $\frac{d}{dx} \text{cosec}(x) = -\text{cosec}(x)\cot(x)$ | $\frac{d}{dx} \text{cosech}(x) = -\text{cosech}(x)\coth(x)$ |
| $\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$ | $\frac{d}{dx} \text{sech}(x) = -\text{sech}(x)\tanh(x)$ |
| Inverse trigonometry function | Inverse hyperbolic function |
|---|---|
| $\frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1 - x^{2}}}$ | $\frac{d}{dx} \sinh^{-1}(x) = \frac{1}{\sqrt{1 + x^{2}}}$ |
| $\frac{d}{dx} \cos^{-1}(x) = \frac{-1}{\sqrt{1 - x^{2}}}$ | $\frac{d}{dx} \cosh^{-1}(x) = \frac{1}{x^{2} - 1}$ |
| $\frac{d}{dx} \tan^{-1}(x) = \frac{1}{1 + x^{2}}$ | $\frac{d}{dx} \tanh^{-1}(x) = \frac{1}{1 - x^{2}}$ |
| $\frac{d}{dx} \cot^{-1}(x) = \frac{-1}{x^{2} + 1}$ | $\frac{d}{dx} \coth^{-1}(x) = \frac{-1}{x^{2} - 1}$ |
| $\frac{d}{dx} \sec^{-1}(x) = \frac{1}{x \sqrt{x^{2} - 1}}$ | $\frac{d}{dx} \text{sech}^{-1}(x) = \frac{-1}{x \sqrt{1 - x^{2}}}$ |
| $\frac{d}{dx} \text{cosec}^{-1}(x) = \frac{-1}{x\sqrt{x^{2} - 1}}$ | $\frac{d}{dx} \text{cosech}^{-1}(x) = \frac{-1}{x \sqrt{x^{2} + 1}}$ |
Find the mnemonic for the derivative of inverse hyperbolic functions at here.
Vector Derivative
Let $\mathbf{x}, \mathbf{u}$ be vectors of length $n$, and let $A$ be a matrix of size $n\times n$ then,
$$ \begin{align} \frac{\partial}{\partial \mathbf{x}} (\mathbf{u}^{T}\mathbf{x}) & = \frac{\partial}{\partial \mathbf{x}} (\mathbf{x}^{T}\mathbf{u}) = \mathbf{u}^{T} \\ \frac{\partial}{\partial \mathbf{x}} (\mathbf{A}\mathbf{x}) & = \mathbf{A} \\ \frac{\partial}{\partial \mathbf{x}} (\mathbf{x}^{T} \mathbf{A} \mathbf{x}) & = \mathbf{x}^{T} (\mathbf{A} + \mathbf{A}^{T}) \\ \frac{\partial^{2}}{\partial \mathbf{x}^{2}} (\mathbf{x^{T}}\mathbf{A}\mathbf{x}) & = \mathbf{A} + \mathbf{A}^{T} \end{align} $$
Note that if $\mathbf{A}$ is a symmetric matrix then, $\mathbf{A} + \mathbf{A}^{T} = 2 \mathbf{A}$.
Integration formulae
Gamma function
- $\forall n \in \mathbb{Z}^{+}$, $\Gamma(n) = (n-1)!$
- Def. $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt, \quad \Re(z)>0$.
Permalink at https://www.physicslog.com/math-notes/calculus
Published on May 7, 2021
Last revised on Jul 2, 2023
