gdzie jesteś: Start / Pochodne / Pochodne funkcji trygonometrycznych / Pochodna funkcji x((lnx)^2)-2xlnx+2x

Pochodna funkcji x((lnx)^2)-2xlnx+2x

$f\left(x\right) =$	$x{\cdot}{\left(\ln\left(x\right)\right)}^{2}-2x{\cdot}\ln\left(x\right)+2x$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x{\cdot}{\left(\ln\left(x\right)\right)}^{2}-2x{\cdot}\ln\left(x\right)+2x\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x{\cdot}{\left(\ln\left(x\right)\right)}^{2}\right)}}-2{\cdot}\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x{\cdot}\ln\left(x\right)\right)}}+2}}$ $=-2{\cdot}\left(\class{steps-node}{\cssId{steps-node-10}{\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}+\class{steps-node}{\cssId{steps-node-12}{x{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}\right)+\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}{\left(\ln\left(x\right)\right)}^{2}}}+\class{steps-node}{\cssId{steps-node-8}{x{\cdot}\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\ln\left(x\right)\right)}^{2}\right)}}}}+2$ $=\class{steps-node}{\cssId{steps-node-15}{1}}{\cdot}{\left(\ln\left(x\right)\right)}^{2}-2{\cdot}\left(\class{steps-node}{\cssId{steps-node-13}{1}}{\cdot}\ln\left(x\right)+\class{steps-node}{\cssId{steps-node-14}{\dfrac{1}{x}}}{\cdot}x\right)+\class{steps-node}{\cssId{steps-node-16}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-17}{\ln\left(x\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-18}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}{\cdot}x+2$ $={\left(\ln\left(x\right)\right)}^{2}-2{\cdot}\left(\ln\left(x\right)+1\right)+2{\cdot}\class{steps-node}{\cssId{steps-node-19}{\dfrac{1}{x}}}{\cdot}x{\cdot}\ln\left(x\right)+2$ $={\left(\ln\left(x\right)\right)}^{2}-2{\cdot}\left(\ln\left(x\right)+1\right)+2{\cdot}\ln\left(x\right)+2$ Uproszczony wynik: $={\left(\ln\left(x\right)\right)}^{2}$

$f\left(x\right) =$

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

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

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

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

$=-2{\cdot}\left(\class{steps-node}{\cssId{steps-node-10}{\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}+\class{steps-node}{\cssId{steps-node-12}{x{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}\right)+\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}{\left(\ln\left(x\right)\right)}^{2}}}+\class{steps-node}{\cssId{steps-node-8}{x{\cdot}\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\ln\left(x\right)\right)}^{2}\right)}}}}+2$

$=\class{steps-node}{\cssId{steps-node-15}{1}}{\cdot}{\left(\ln\left(x\right)\right)}^{2}-2{\cdot}\left(\class{steps-node}{\cssId{steps-node-13}{1}}{\cdot}\ln\left(x\right)+\class{steps-node}{\cssId{steps-node-14}{\dfrac{1}{x}}}{\cdot}x\right)+\class{steps-node}{\cssId{steps-node-16}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-17}{\ln\left(x\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-18}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}{\cdot}x+2$

$={\left(\ln\left(x\right)\right)}^{2}-2{\cdot}\left(\ln\left(x\right)+1\right)+2{\cdot}\class{steps-node}{\cssId{steps-node-19}{\dfrac{1}{x}}}{\cdot}x{\cdot}\ln\left(x\right)+2$

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

Uproszczony wynik:

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