User:Tikai/int

laws of sines and cosines
$$\sin (\theta+\phi)=\cos\phi\sin\theta+\sin\phi\cos\theta\!$$

$$\cos (\theta+\phi)=\cos\phi\cos\theta-\sin\phi\sin\theta\!$$

basics
$$\cfrac{d(uv)}{dx}=u\cfrac{dv}{x}+v\cfrac{du}{x}$$

$$\cfrac{dw}{dx}=\cfrac{dw}{du}\cfrac{du}{dx}$$

$$\cfrac{d\cfrac{u}{v}}{dx}=\cfrac{v\cfrac{du}{dx}-u\cfrac{dv}{dx}}{v^2}$$

sines and cosines
$$\cfrac{d\sin\theta}{d\theta}=\cos\theta$$

$$\cfrac{d\cos\theta}{d\theta}=-\sin\theta$$

logarithm and powers
$$e=2.71828...\,$$

$$\ln x = \log_e x\,$$

$$\log_e x=\cfrac{\log_{10} x}{\log_{10} e}\rightarrow\log_e x=2.303\log_{10} x$$

$$\cfrac{d\ln x}{dx}=\frac{1}{x}$$

$$y=a^x\,$$

$$\frac{d\ln y}{dx}=\frac{dy}{dx}\frac{d\ln y}{dy}=\frac{1}{y}\frac{da^x}{dx}=\frac{x\ln a}{dx}=\frac{dx}{dx}\ln a+\frac{d\ln a}{dx}x=\ln a\Rightarrow\frac{da^x}{dx}=a^x\ln a \Rightarrow\frac{de^x}{dx}=e^x$$