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Pochodna funkcji (1-lnx)/x^2

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

$f\left(x\right) =$

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

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

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

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

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

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

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

Wynik alternatywny:

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