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Pochodna funkcji 2xlnx+x

$f\left(x\right) =$	$2x{\cdot}\ln\left(x\right)+x$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(2x{\cdot}\ln\left(x\right)+x\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{2{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x{\cdot}\ln\left(x\right)\right)}}+1}}$ $=2{\cdot}\left(\class{steps-node}{\cssId{steps-node-5}{\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}+\class{steps-node}{\cssId{steps-node-7}{x{\cdot}\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}\right)+1$ $=2{\cdot}\left(\class{steps-node}{\cssId{steps-node-8}{1}}{\cdot}\ln\left(x\right)+\class{steps-node}{\cssId{steps-node-9}{\dfrac{1}{x}}}{\cdot}x\right)+1$ $=2{\cdot}\left(\ln\left(x\right)+1\right)+1$ Uproszczony wynik: $=2{\cdot}\ln\left(x\right)+3$

$f\left(x\right) =$

$2x{\cdot}\ln\left(x\right)+x$

$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(2x{\cdot}\ln\left(x\right)+x\right)}}$

$=\class{steps-node}{\cssId{steps-node-2}{2{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x{\cdot}\ln\left(x\right)\right)}}+1}}$

$=2{\cdot}\left(\class{steps-node}{\cssId{steps-node-5}{\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}+\class{steps-node}{\cssId{steps-node-7}{x{\cdot}\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}\right)+1$

$=2{\cdot}\left(\class{steps-node}{\cssId{steps-node-8}{1}}{\cdot}\ln\left(x\right)+\class{steps-node}{\cssId{steps-node-9}{\dfrac{1}{x}}}{\cdot}x\right)+1$

$=2{\cdot}\left(\ln\left(x\right)+1\right)+1$

Uproszczony wynik:

$=2{\cdot}\ln\left(x\right)+3$