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Pochodna funkcji (-2x^2+2x)/((x-1)^2)

$f\left(x\right) =$	$\dfrac{2x-2{x}^{2}}{{\left(x-1\right)}^{2}}$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{2x-2{x}^{2}}{{\left(x-1\right)}^{2}}\right)}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-4}{{\left(x-1\right)}^{2}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(2x-2{x}^{2}\right)}}}}-\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(x-1\right)}^{2}\right)}}{\cdot}\left(2x-2{x}^{2}\right)}}}{\class{steps-node}{\cssId{steps-node-2}{{\left({\left(x-1\right)}^{2}\right)}^{2}}}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-7}{\left(2-2{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}\right)}}{\cdot}{\left(x-1\right)}^{2}-\class{steps-node}{\cssId{steps-node-9}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-10}{\left(x-1\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}{\cdot}\left(2x-2{x}^{2}\right)}{{\left(x-1\right)}^{4}}$ $=\dfrac{\left(2-2{\cdot}\class{steps-node}{\cssId{steps-node-12}{2}}\class{steps-node}{\cssId{steps-node-13}{x}}\right){\cdot}{\left(x-1\right)}^{2}-2{\cdot}\class{steps-node}{\cssId{steps-node-14}{1}}{\cdot}\left(x-1\right){\cdot}\left(2x-2{x}^{2}\right)}{{\left(x-1\right)}^{4}}$ $=\dfrac{\left(2-4x\right){\cdot}{\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}\left(2x-2{x}^{2}\right)}{{\left(x-1\right)}^{4}}$ Uproszczony wynik: $=\dfrac{2-4x}{{\left(x-1\right)}^{2}}-\dfrac{2{\cdot}\left(2x-2{x}^{2}\right)}{{\left(x-1\right)}^{3}}$

$f\left(x\right) =$

$\dfrac{2x-2{x}^{2}}{{\left(x-1\right)}^{2}}$

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-4}{{\left(x-1\right)}^{2}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(2x-2{x}^{2}\right)}}}}-\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(x-1\right)}^{2}\right)}}{\cdot}\left(2x-2{x}^{2}\right)}}}{\class{steps-node}{\cssId{steps-node-2}{{\left({\left(x-1\right)}^{2}\right)}^{2}}}}$

$=\dfrac{\class{steps-node}{\cssId{steps-node-7}{\left(2-2{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}\right)}}{\cdot}{\left(x-1\right)}^{2}-\class{steps-node}{\cssId{steps-node-9}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-10}{\left(x-1\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}{\cdot}\left(2x-2{x}^{2}\right)}{{\left(x-1\right)}^{4}}$

$=\dfrac{\left(2-2{\cdot}\class{steps-node}{\cssId{steps-node-12}{2}}\class{steps-node}{\cssId{steps-node-13}{x}}\right){\cdot}{\left(x-1\right)}^{2}-2{\cdot}\class{steps-node}{\cssId{steps-node-14}{1}}{\cdot}\left(x-1\right){\cdot}\left(2x-2{x}^{2}\right)}{{\left(x-1\right)}^{4}}$

$=\dfrac{\left(2-4x\right){\cdot}{\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}\left(2x-2{x}^{2}\right)}{{\left(x-1\right)}^{4}}$

Uproszczony wynik:

$=\dfrac{2-4x}{{\left(x-1\right)}^{2}}-\dfrac{2{\cdot}\left(2x-2{x}^{2}\right)}{{\left(x-1\right)}^{3}}$