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Pochodna funkcji (1/x)^(1/x)

$f\left(x\right) =$	$\dfrac{1}{{x}^{\frac{1}{x}}}$ Note: Your input has been rewritten/simplified.
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{1}{{x}^{\frac{1}{x}}}\right)}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-4}{-\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{\frac{1}{x}}\right)}}}}}{\class{steps-node}{\cssId{steps-node-2}{{\left({x}^{\frac{1}{x}}\right)}^{2}}}}$ $=\dfrac{-\left(\class{steps-node}{\cssId{steps-node-5}{{x}^{\frac{1}{x}}}}{\cdot}\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right){\cdot}\dfrac{1}{x}\right)}}\right)}{{x}^{\frac{2}{x}}}$ $=\dfrac{-\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{\ln\left(x\right)}{x}\right)}}}{{x}^{\frac{1}{x}}}$ $=\dfrac{-\dfrac{\class{steps-node}{\cssId{steps-node-10}{x{\cdot}\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}-\class{steps-node}{\cssId{steps-node-12}{\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}}{\class{steps-node}{\cssId{steps-node-8}{{x}^{2}}}}}{{x}^{\frac{1}{x}}}$ $=-{x}^{\frac{-1}{x}-2}{\cdot}\left(\class{steps-node}{\cssId{steps-node-13}{\dfrac{1}{x}}}{\cdot}x-\class{steps-node}{\cssId{steps-node-14}{1}}{\cdot}\ln\left(x\right)\right)$ $=-{x}^{\frac{-1}{x}-2}{\cdot}\left(1-\ln\left(x\right)\right)$ Wynik alternatywny: $=\dfrac{\dfrac{\ln\left(x\right)}{{x}^{2}}-\dfrac{1}{{x}^{2}}}{{x}^{\frac{1}{x}}}$

$f\left(x\right) =$

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

Note: Your input has been rewritten/simplified.

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

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

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

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

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

$=\dfrac{-\dfrac{\class{steps-node}{\cssId{steps-node-10}{x{\cdot}\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}-\class{steps-node}{\cssId{steps-node-12}{\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}}{\class{steps-node}{\cssId{steps-node-8}{{x}^{2}}}}}{{x}^{\frac{1}{x}}}$

$=-{x}^{\frac{-1}{x}-2}{\cdot}\left(\class{steps-node}{\cssId{steps-node-13}{\dfrac{1}{x}}}{\cdot}x-\class{steps-node}{\cssId{steps-node-14}{1}}{\cdot}\ln\left(x\right)\right)$

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

Wynik alternatywny:

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