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Pochodna funkcji ((x-1)^2-2x(x-1))/((x-1)^4)

$f\left(x\right) =$	$\dfrac{{\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x}{{\left(x-1\right)}^{4}}$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{{\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x}{{\left(x-1\right)}^{4}}\right)}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-4}{{\left(x-1\right)}^{4}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}}}}-\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(x-1\right)}^{4}\right)}}{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}}}{\class{steps-node}{\cssId{steps-node-2}{{\left({\left(x-1\right)}^{4}\right)}^{2}}}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-7}{\left(\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(x-1\right)}^{2}\right)}}-2{\cdot}\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\left(x-1\right){\cdot}x\right)}}\right)}}{\cdot}{\left(x-1\right)}^{4}-\class{steps-node}{\cssId{steps-node-10}{4}}{\cdot}\class{steps-node}{\cssId{steps-node-11}{{\left(x-1\right)}^{3}}}{\cdot}\class{steps-node}{\cssId{steps-node-12}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{8}}$ $=\dfrac{\left(\class{steps-node}{\cssId{steps-node-13}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-14}{\left(x-1\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-15}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}-2{\cdot}\left(\class{steps-node}{\cssId{steps-node-17}{\class{steps-node}{\cssId{steps-node-16}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}{\cdot}x}}+\class{steps-node}{\cssId{steps-node-19}{\left(x-1\right){\cdot}\class{steps-node}{\cssId{steps-node-18}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}\right)\right){\cdot}{\left(x-1\right)}^{4}-4{\cdot}\class{steps-node}{\cssId{steps-node-20}{1}}{\cdot}{\left(x-1\right)}^{3}{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{8}}$ $=\dfrac{{\left(x-1\right)}^{4}{\cdot}\left(2{\cdot}\class{steps-node}{\cssId{steps-node-21}{1}}{\cdot}\left(x-1\right)-2{\cdot}\left(\class{steps-node}{\cssId{steps-node-22}{1}}x+\class{steps-node}{\cssId{steps-node-23}{1}}{\cdot}\left(x-1\right)\right)\right)-4{\cdot}{\left(x-1\right)}^{3}{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{8}}$ $=\dfrac{{\left(x-1\right)}^{4}{\cdot}\left(2{\cdot}\left(x-1\right)-2{\cdot}\left(2x-1\right)\right)-4{\cdot}{\left(x-1\right)}^{3}{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{8}}$ Uproszczony wynik: $=\dfrac{-4{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{5}}-\dfrac{2x}{{\left(x-1\right)}^{4}}$

$f\left(x\right) =$

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

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

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

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

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

$=\dfrac{\left(\class{steps-node}{\cssId{steps-node-13}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-14}{\left(x-1\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-15}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}-2{\cdot}\left(\class{steps-node}{\cssId{steps-node-17}{\class{steps-node}{\cssId{steps-node-16}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}{\cdot}x}}+\class{steps-node}{\cssId{steps-node-19}{\left(x-1\right){\cdot}\class{steps-node}{\cssId{steps-node-18}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}\right)\right){\cdot}{\left(x-1\right)}^{4}-4{\cdot}\class{steps-node}{\cssId{steps-node-20}{1}}{\cdot}{\left(x-1\right)}^{3}{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{8}}$

$=\dfrac{{\left(x-1\right)}^{4}{\cdot}\left(2{\cdot}\class{steps-node}{\cssId{steps-node-21}{1}}{\cdot}\left(x-1\right)-2{\cdot}\left(\class{steps-node}{\cssId{steps-node-22}{1}}x+\class{steps-node}{\cssId{steps-node-23}{1}}{\cdot}\left(x-1\right)\right)\right)-4{\cdot}{\left(x-1\right)}^{3}{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{8}}$

$=\dfrac{{\left(x-1\right)}^{4}{\cdot}\left(2{\cdot}\left(x-1\right)-2{\cdot}\left(2x-1\right)\right)-4{\cdot}{\left(x-1\right)}^{3}{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{8}}$

Uproszczony wynik:

$=\dfrac{-4{\cdot}\left({\left(x-1\right)}^{2}-2{\cdot}\left(x-1\right){\cdot}x\right)}{{\left(x-1\right)}^{5}}-\dfrac{2x}{{\left(x-1\right)}^{4}}$