Derivatives

1. ddx[af(x)]=addxf(x)$\frac{d}{dx}\left[af\left(x\right)\right]=a\frac{d}{dx}f\left(x\right)$
2. ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$
3. ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)$\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}g\left(x\right)+g\left(x\right)\frac{d}{dx}f\left(x\right)$
4. ddxf(u)=[dduf(u)]dudx$\frac{d}{dx}f\left(u\right)=\left[\frac{d}{du}f\left(u\right)\right]\frac{du}{dx}$
5. ddxxm=mxm−1$\frac{d}{dx}{x}^m=m{x}^{m-1}$
6. ddxsinx=cosx$\frac{d}{dx}\text{sin}x=\text{cos}x$
7. ddxcosx=−sinx$\frac{d}{dx}\text{cos}x=\text{−}\text{sin}x$
8. ddxtanx=sec2x$\frac{d}{dx}\text{tan}x={\text{sec}}^{2}x$
9. ddxcotx=−csc2x$\frac{d}{dx}\text{cot}x=\text{−}{\text{csc}}^{2}x$
10. ddxsecx=tanxsecx$\frac{d}{dx}\text{sec}x=\text{tan}x\text{sec}x$
11. ddxcscx=−cotxcscx$\frac{d}{dx}\text{csc}x=\text{−}\text{cot}x\text{csc}x$
12. ddxex=ex$\frac{d}{dx}{e}^x=e^x$
13. ddxlnx=1x$\frac{d}{dx}\text{ln}x=\frac{1}{x}$
14. ddxsin−1x=11−x2√$\frac{d}{dx}{\text{sin}}^{\mathrm{-1}}x=\frac{1}{\sqrt{1-x^2}}$
15. ddxcos−1x=−11−x2√$\frac{d}{dx}{\text{cos}}^{\mathrm{-1}}x=-\frac{1}{\sqrt{1-x^2}}$
16. ddxtan−1x=11+x2