# Say f(x)=ln(1+2x+2x^2) or g(x)=tan(2x^(4)−x).Using the definition leads to messy derivatives almost immediately.If it was some simple rational function,for example,i would try to use Maclaurin Series of 1/1+x or 1/1−x . Think of any shortcut for listed above functions.

Hope Hancock 2022-10-03 Answered
Say$f\left(x\right)=\mathrm{ln}\left(1+2x+2{x}^{2}\right)$ or $g\left(x\right)=\mathrm{tan}\left(2{x}^{4}-x\right)$.Using the definition leads to messy derivatives almost immediately.If it was some simple rational function,for example,i would try to use Maclaurin Series of $\frac{1}{1+x}$ or $\frac{1}{1-x}$ and then manipulate it to get result
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trutdelamodej0
keep substitution in mind. It may not give the whole infinite series, but you will usually get the first several terms. So,
$\frac{1}{1+t}=1-t+{t}^{2}-{t}^{3}+{t}^{4}-{t}^{5}\cdots$
$\mathrm{log}\left(1+t\right)=t-\frac{{t}^{2}}{2}+\frac{{t}^{3}}{3}-\frac{{t}^{4}}{4}\cdots$
Taking $t=2x+2{x}^{2}$ correctly gives the first few terms of $\mathrm{log}\left(1+2x+2{x}^{2}\right),$, up to ${x}^{4}$
$\mathrm{log}\left(1+2x+2{x}^{2}\right)=2x-\frac{4{x}^{3}}{3}+2{x}^{4}\cdots$