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a)
D\left(e^{x^3-1}\right)=3x^2e^{x^3-1}
b)
D\ \frac{e^x}{x}=\frac{e^x\cdot x-1\cdot e^x}{x^2}=\frac{e^x\left(x-1\right)}{x^2}
c)
 
D\ \frac{x+1}{e^x}=\frac{1\cdot e^{2x}-2e^{2x}\cdot\left(x+1\right)}{e^{2x}}==-x+e^{-x}
d)
D\frac{2}{e^{2x}}=2e^{-2x}=-4e^{-2x}
e)
D\left(x^2e^{2x}\right)=2x\cdot e^{2x}+x^2\cdot e^{2x}=e^{2x}\left(x^2+2x\right)
f)
D\left(\frac{e^x+x}{x}\right)=\frac{x\left(e^x+1\right)-1\left(e^x+x\right)}{x^2}=\frac{\left(x-1\right)\left(e^x+x\right)}{x^2}