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Pochodna funkcji x^(1/x)

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

$f\left(x\right) =$

${x}^{\frac{1}{x}}$

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

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

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-7}{x{\cdot}\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}-\class{steps-node}{\cssId{steps-node-9}{\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}}{\class{steps-node}{\cssId{steps-node-5}{{x}^{2}}}}{\cdot}{x}^{\frac{1}{x}}$

$={x}^{\frac{1}{x}-2}{\cdot}\left(\class{steps-node}{\cssId{steps-node-10}{\dfrac{1}{x}}}{\cdot}x-\class{steps-node}{\cssId{steps-node-11}{1}}{\cdot}\ln\left(x\right)\right)$

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

Wynik alternatywny:

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