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Pochodna funkcji (3(lnx)^2)/x

$f\left(x\right) =$	$\dfrac{3{\cdot}{\left(\ln\left(x\right)\right)}^{2}}{x}$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{3{\cdot}{\left(\ln\left(x\right)\right)}^{2}}{x}\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{3{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{{\left(\ln\left(x\right)\right)}^{2}}{x}\right)}}}}$ $=3{\cdot}\dfrac{\class{steps-node}{\cssId{steps-node-6}{x{\cdot}\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\ln\left(x\right)\right)}^{2}\right)}}}}-\class{steps-node}{\cssId{steps-node-8}{\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}{\left(\ln\left(x\right)\right)}^{2}}}}{\class{steps-node}{\cssId{steps-node-4}{{x}^{2}}}}$ $=\dfrac{3{\cdot}\left(\class{steps-node}{\cssId{steps-node-9}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-10}{\ln\left(x\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}{\cdot}x-\class{steps-node}{\cssId{steps-node-12}{1}}{\cdot}{\left(\ln\left(x\right)\right)}^{2}\right)}{{x}^{2}}$ $=\dfrac{3{\cdot}\left(2{\cdot}\class{steps-node}{\cssId{steps-node-13}{\dfrac{1}{x}}}{\cdot}x{\cdot}\ln\left(x\right)-{\left(\ln\left(x\right)\right)}^{2}\right)}{{x}^{2}}$ $=\dfrac{3{\cdot}\left(2{\cdot}\ln\left(x\right)-{\left(\ln\left(x\right)\right)}^{2}\right)}{{x}^{2}}$ Wynik alternatywny: $=\dfrac{6{\cdot}\ln\left(x\right)}{{x}^{2}}-\dfrac{3{\cdot}{\left(\ln\left(x\right)\right)}^{2}}{{x}^{2}}$

$f\left(x\right) =$

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

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

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

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

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

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

$=\dfrac{3{\cdot}\left(2{\cdot}\class{steps-node}{\cssId{steps-node-13}{\dfrac{1}{x}}}{\cdot}x{\cdot}\ln\left(x\right)-{\left(\ln\left(x\right)\right)}^{2}\right)}{{x}^{2}}$

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

Wynik alternatywny:

$=\dfrac{6{\cdot}\ln\left(x\right)}{{x}^{2}}-\dfrac{3{\cdot}{\left(\ln\left(x\right)\right)}^{2}}{{x}^{2}}$