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

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

$f\left(x\right) =$

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

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

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

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

$=\class{steps-node}{\cssId{steps-node-6}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-7}{\left(1-x\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(1-x\right)}}{\cdot}{x}^{2}+\class{steps-node}{\cssId{steps-node-9}{2}}\class{steps-node}{\cssId{steps-node-10}{x}}{\cdot}{\left(1-x\right)}^{2}$

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

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

Uproszczony wynik:

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