Calculus AB Derivation/Integration Euler/Log + Inverse Identities

d/dx $\ln\left(f\left(x\right)\right)$

$$\frac{f^{\prime}\left(x\right)}{f\left(x\right)}$$

The derivative of a general inverse function, where f(a)=b and f^-1(b) = g(b) = a

$$g^{\prime}\left(b\right)=\frac{1}{f^{\prime}\left(a\right)}$$

$$\int_{}^{}\!g^{\prime}\left(x\right)\,e^{g\left(x\right)}\cdot dx$$

$$e^{g\left(x\right)}+C$$

d/dx ($\log_{b}x$)

$$\frac{1}{\ln b}\cdot\frac{g^{\prime}\left(x\right)}{g\left(x\right)}$$

d/dx ($a^{x}$)

$$a^{x}\cdot\ln a$$

$$\int a^{x}dx$$

$$\frac{a^{x}}{\ln a}+c$$

d/dx ($\sin^{-1}\left(u\right)$)

$$\frac{u^{\prime}}{\sqrt{1-u^2}}$$

d/dx ($\cos^{-1}\left(u\right)$ )

$$-\frac{u^{\prime}}{\sqrt{1-u^2}}$$

d/dx ($\tan^{-1}\left(u\right)$ )

$$\frac{u^{\prime}}{u^2+1}$$

d/dx ($\cot^{-1}\left(u\right)$)

$$-\frac{u^{\prime}}{u^2+1}$$

d/dx ($\sec^{-1}\left(u\right)$)

$$\frac{u^{\prime}}{\left|u\right|\sqrt{u^2-1}}$$

d/dx ($\csc^{-1}\left(u\right)$)

$$-\frac{u^{\prime}}{\left|u\right|\sqrt{u^2-1}}$$

$$\int\frac{1}{\sqrt{\left(a^2-u^2\right)}}du$$

$$\arcsin\left(\frac{u}{a}\right)+C$$

$$\int\frac{1}{u^2+a^2}du$$

$$\frac{1}{a}\arctan\left(\frac{u}{a}\right)+C$$

$$\int\frac{1}{u\sqrt{u^2-a^2}}du$$

$$\frac{1}{a}\operatorname{arcsec}\left(\frac{\left|u\right|}{a}\right)+C$$