Derivatives, Integrals, and Properties of Inverse Trigonometric and Hyperbolic Functions

$$\underset{k}{\LARGE\Delta} a_k \qquad := \qquad 1 + a_1 + a_1\cdot a_2 + a_1\cdot a_2 \cdot a_3 + \dots\qquad$$

$$\frac{1}{1-x} \qquad = \qquad \underset{k}{\LARGE\Delta} x \qquad (|x| < 1)$$

$$\left(\frac{1}{1-x}\right)^{m+1} \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{k+m}{k} x $$

$$\left(\frac{1}{1-x}\right)^{m+1} \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{k-m-2}{k} \left( \frac{x}{x-1} \right) $$

$$\mathrm{e}^x \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{1}{k}x \qquad (|x| < \infty)$$

$$\sin x \qquad = \qquad x\underset{k}{\LARGE\Delta} \frac{-1}{(1+2k)^{\underline{2}}}x^2 \qquad (|x| < \infty)$$

$$\cos x \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{-1}{(2k)^{\underline{2}}}x^2 \qquad (|x| < \infty)$$

$$\ln x \qquad = \qquad x\underset{k}{\LARGE\Delta} \frac{-k}{k+1}x \qquad (-1 < |x| \le 1)$$

$$\ln x \qquad = \qquad 2\left(\frac{x-1}{x+1}\right)\underset{k}{\LARGE\Delta} \frac{2k-1}{2k+1}\left( \frac{x-1}{x+1} \right)^2 \qquad $$

$$(1+x)^n \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{n+1-k}{k}x \qquad (|x| < 1)$$

$$\tan^{-1} x \qquad = \qquad x\underset{k}{\LARGE\Delta} \frac{1-2k}{1+2k}x^2 \qquad (|x| < 1)$$

$$\tan^{-1} x \qquad = \qquad \frac{\pi}{2} - \frac{1}{x}\underset{k}{\LARGE\Delta} \frac{1-2k}{(1+2k)x^2} \qquad (x \ge 1)$$

$$\tan^{-1} x \qquad = \qquad -\frac{\pi}{2} - \frac{1}{x}\underset{k}{\LARGE\Delta} \frac{1-2k}{(1+2k)x^2} \qquad (x \le -1)$$

$$\sin^{-1} x \qquad = \qquad x\underset{k}{\LARGE\Delta} \frac{(1-2k)^{2}}{(1+2k)^{\underline{2}}}x^2 \qquad (|x| < \infty)$$

Normal Distribution

$$ \operatorname{P}(a < x < b) \qquad = \qquad \frac{x}{\sqrt{2\pi}}\underset{k}{\LARGE\Delta} \frac{1-2k}{(1+2k)^{\underline{2}}}x^2 \Big|_a^b$$

$$ \operatorname{erf}(x) \qquad = \qquad \frac{2x}{\sqrt{\pi}}\underset{k}{\LARGE\Delta} \frac{1-2k}{(1+2k)k}x^2 $$

$$ \operatorname{\beta}_x(a, b) \qquad = \qquad \frac{(1-x)^b}{b}\underset{k}{\LARGE\Delta} \frac{(k-a)(k+b-1)}{(k+b)k}(1-x) \qquad \text{if} \quad x > \frac{a+1}{a+b+2}$$

$$ \operatorname{\beta}_x(a, b) \qquad = \qquad \frac{x^a}{a}\underset{k}{\LARGE\Delta} \frac{(k+a-1)(k-b)}{(k+a)k}x \qquad \text{if} \quad x < \frac{a+1}{a+b+2}$$

Student T Distribution

$$ \operatorname{P}(x < t) \qquad = \qquad \frac{1}{2} + \frac{t}{\pi\sqrt{t^2 + v}}\left(\underset{k}{\LARGE\Delta} \frac{k+v-2}{2k+v-1} \right)\left( \underset{k}{\LARGE\Delta}\frac{(2k-1)(2k-v)}{(2k+1)^{\underline{2}}}\cdot \frac{t^2}{t^2+v}\right)$$

Generalized Binomial

$$ (x+1)^\alpha \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{\alpha + 1-k}{k} x $$

$$ x^\frac{n}{m} \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{n+m-km}{km} (x-1) $$

Example Functions

$$ \sqrt{1-x} \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{2k-3}{2k} x $$

$$ (1+x)^n-(1-x)^n \qquad = \qquad 2nx\underset{k}{\LARGE\Delta} \frac{(n+1-2k)^{\underline{2}}}{(1+2k)^{\underline{2}}} x^2 $$

$$ (1+x)^n+(1-x)^n \qquad = \qquad 2\underset{k}{\LARGE\Delta} \frac{(n+2-2k)^{\underline{2}}}{(2k)^{\underline{2}}} x^2 $$

$$ \sqrt{1-x^2} \quad = \quad \cos(\sin^{-1} x) \quad = \quad \underset{k}{\LARGE\Delta} \frac{2k-3}{2k} x^2 $$

$$ \cos x - \sin x \quad = \quad \sqrt{2}\cos(x-\frac{\pi}{4}) \quad = \quad \underset{k}{\LARGE\Delta} \frac{(-1)^k}{k} x $$

$$ \cos x + \sin x \quad = \quad \sqrt{2}\cos(x+\frac{\pi}{4}) \quad = \quad \underset{k}{\LARGE\Delta} \frac{(-1)^{k+1}}{k} x $$

$$ \frac{\sin^{-1} x}{\sqrt{1-x^2}} \quad = \quad x\underset{k}{\LARGE\Delta} \frac{2k}{2k+1} x^2 $$

Example Constants

$$ \frac{\pi}{3} \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{(1-2k)^2}{8k(2k+1)} \quad \left( \text{from} \sin^{-1}\left(\frac{1}{2}\right) \right)$$

$$ \frac{4}{3}+\frac{2\pi \sqrt{3}}{27} \qquad = \qquad \underset{k}{\LARGE\Delta} \frac{k}{2(2k-1)} $$

Properties

$$ \frac{d}{dx}\left( \underset{k}{\LARGE\Delta} a_k x \right) \quad = \quad a_1 \underset{k}{\LARGE\Delta} \frac{1+k}{k}a_{1+k} x$$

$$ \frac{d}{dx}\left( \underset{k}{\LARGE\Delta} a_k x^n \right) \quad = \quad n a_1 x^{n-1}\underset{k}{\LARGE\Delta} \frac{1+k}{k}a_{1+k} x^n$$

$$ \frac{d}{dx}\left[ x\underset{k}{\LARGE\Delta} a_k x^2 \right] \quad = \quad \underset{k}{\LARGE\Delta} \frac{2k+1}{2k-1}a_{k} x^2$$

$$ \int\left( \underset{k}{\LARGE\Delta} a_k x \right)dx \quad = \quad x\underset{k}{\LARGE\Delta} \frac{k}{1+k}a_{k} x$$

$$ \int\left( \underset{k}{\LARGE\Delta} a_k x^n \right)dx \quad = \quad x\underset{k}{\LARGE\Delta} \frac{(k-1)n + 1}{kn+1}a_{k} x^n$$

$$ \int\left( \underset{k}{\LARGE\Delta} a_k x^2 \right)dx \quad = \quad x\underset{k}{\LARGE\Delta} \frac{2k-1}{2k+1}a_{k} x^2$$

$$ \operatorname{EVEN}\left( \underset{k}{\LARGE\Delta} a_k x \right) \quad = \quad \underset{k}{\LARGE\Delta} a_{2k}a_{2k-1} x^2$$

$$ \operatorname{ODD}\left( \underset{k}{\LARGE\Delta} a_k x \right) \quad = \quad a_1 x\underset{k}{\LARGE\Delta} a_{2k+1}a_{2k} x^2$$

$$ \Large\sum_{z=0}^{\infty} a_z \quad = \quad a_0 \underset{k}{\LARGE\Delta} \frac{a_k}{a_{k-1}}$$

Sterling Number $$ \left\{k \atop m \right\} = m \left\{k-1 \atop m \right\} + \left\{k-1 \atop m-1 \right\}$$

$$ \left\{k \atop m \right\} \quad = \quad \frac{(-1)^{m+1}}{(m-1)^\underline{m-1}} \underset{k}{\LARGE\Delta} \frac{(z-m)(z+1)^{k-1}}{z^k}$$

Lambert W Function $$ \left( \operatorname{W}(z)e^{\operatorname{W}(z)} = z\right)$$

$$ \operatorname{W}(z) \quad = \quad z\underset{k}{\LARGE\Delta} \left(\frac{k+1}{k}\right)^{k-1}(-z) \qquad |z| < \frac{1}{e} $$

Taylor Series as Term Ratio