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

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

$f\left(x\right) =$

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

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

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

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

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

$=\dfrac{{\left(x-1\right)}^{2}-2{\cdot}\class{steps-node}{\cssId{steps-node-11}{1}}{\cdot}\left(x-1\right){\cdot}x}{{\left(x-1\right)}^{4}}$

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

Uproszczony wynik:

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