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Pochodna funkcji lnx/x

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

$f\left(x\right) =$

$\dfrac{\ln\left(x\right)}{x}$

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

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-7}{\dfrac{1}{x}}}{\cdot}x-\class{steps-node}{\cssId{steps-node-8}{1}}{\cdot}\ln\left(x\right)}{{x}^{2}}$

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

Wynik alternatywny:

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