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Pochodna funkcji (a/x)^x

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

$f\left(a, x\right) =$

${\left(\dfrac{a}{x}\right)}^{x}$

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

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

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

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

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

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

$={\left(\dfrac{a}{x}\right)}^{x}{\cdot}\left(-\ln\left(x\right)-\class{steps-node}{\cssId{steps-node-12}{\dfrac{1}{x}}}{\cdot}x+\ln\left(a\right)\right)$

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