Propriedades das Derivadas
Neste post apresentam-se as Propriedades das Derivadas, na qual tornam o processo de derivação mais prático e consequentemente mais rápido.
Sabendo que 
Caso queira ver as demonstrações clique nas propriedades.
a) Derivada de uma constante:
![\displaystyle \frac{d}{dx}\left[c\right]=0](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%5Bc%5Cright%5D%3D0&bg=ffffff&fg=000000&s=1 "\displaystyle \frac{d}{dx}\left[c\right]=0")
b) Derivada do produto de uma constante por uma função:
![\displaystyle \frac{d}{dx}[cf(x)]=c\frac{d}{dx}[f(x)]](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdx%7D%5Bcf%28x%29%5D%3Dc%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D&bg=ffffff&fg=000000&s=1 "\displaystyle \frac{d}{dx}[cf(x)]=c\frac{d}{dx}[f(x)]")
c) Derivada da soma ou subtração:
![\displaystyle \frac{d}{dx}[f(x)\pm g(x)]=\frac{d}{dx}[f(x)]\pm\frac{d}{dx}[g(x)]](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5Cpm+g%28x%29%5D%3D%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D%5Cpm%5Cfrac%7Bd%7D%7Bdx%7D%5Bg%28x%29%5D&bg=ffffff&fg=000000&s=1 "\displaystyle \frac{d}{dx}[f(x)\pm g(x)]=\frac{d}{dx}[f(x)]\pm\frac{d}{dx}[g(x)]")
d) Derivada do produto:
![\displaystyle \frac{d}{dx}[f(x)g(x)]=g(x)\frac{d}{dx}[f(x)]+f(x)\frac{d}{dx}[g(x)]](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29g%28x%29%5D%3Dg%28x%29%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D%2Bf%28x%29%5Cfrac%7Bd%7D%7Bdx%7D%5Bg%28x%29%5D&bg=ffffff&fg=000000&s=1 "\displaystyle \frac{d}{dx}[f(x)g(x)]=g(x)\frac{d}{dx}[f(x)]+f(x)\frac{d}{dx}[g(x)]")
e) Derivada da divisão:
![\displaystyle \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)\frac{d}{dx}[f(x)]-f(x)\frac{d}{dx}[g(x)]}{g(x)^{2}}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%5B%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D%5Cright%5D%3D%5Cfrac%7Bg%28x%29%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D-f%28x%29%5Cfrac%7Bd%7D%7Bdx%7D%5Bg%28x%29%5D%7D%7Bg%28x%29%5E%7B2%7D%7D&bg=ffffff&fg=000000&s=1 "\displaystyle \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)\frac{d}{dx}[f(x)]-f(x)\frac{d}{dx}[g(x)]}{g(x)^{2}}")
sendo 
f) Derivada da potência:
![\displaystyle \frac{d}{dx}\big[x^{n}\big]=n\; x^{n-1}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdx%7D%5Cbig%5Bx%5E%7Bn%7D%5Cbig%5D%3Dn%5C%3B+x%5E%7Bn-1%7D&bg=ffffff&fg=000000&s=1 "\displaystyle \frac{d}{dx}\big[x^{n}\big]=n\; x^{n-1}")
g) Regra da cadeia:
![\displaystyle \frac{d}{dx}\bigg[f\big(g(x)\big)\bigg]=\frac{d}{du}\big[f(u)\big]\cdot \frac{d}{dx}\big[g(x)\big]](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdx%7D%5Cbigg%5Bf%5Cbig%28g%28x%29%5Cbig%29%5Cbigg%5D%3D%5Cfrac%7Bd%7D%7Bdu%7D%5Cbig%5Bf%28u%29%5Cbig%5D%5Ccdot+%5Cfrac%7Bd%7D%7Bdx%7D%5Cbig%5Bg%28x%29%5Cbig%5D&bg=ffffff&fg=000000&s=1 "\displaystyle \frac{d}{dx}\bigg[f\big(g(x)\big)\bigg]=\frac{d}{du}\big[f(u)\big]\cdot \frac{d}{dx}\big[g(x)\big]")
onde 
