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Pochodna funkcji ln(ln(ln(x)))

$f\left(x\right) =$	$\ln\left(\ln\left(\ln\left(x\right)\right)\right)$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(\ln\left(\ln\left(x\right)\right)\right)\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{\dfrac{1}{\ln\left(\ln\left(x\right)\right)}}}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(\ln\left(x\right)\right)\right)}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-4}{\dfrac{1}{\ln\left(x\right)}}}{\cdot}\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}{\ln\left(\ln\left(x\right)\right)}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-6}{\dfrac{1}{x}}}}{\ln\left(x\right){\cdot}\ln\left(\ln\left(x\right)\right)}$ $=\dfrac{1}{x{\cdot}\ln\left(x\right){\cdot}\ln\left(\ln\left(x\right)\right)}$

$f\left(x\right) =$

$\ln\left(\ln\left(\ln\left(x\right)\right)\right)$

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

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-4}{\dfrac{1}{\ln\left(x\right)}}}{\cdot}\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}{\ln\left(\ln\left(x\right)\right)}$

$=\dfrac{\class{steps-node}{\cssId{steps-node-6}{\dfrac{1}{x}}}}{\ln\left(x\right){\cdot}\ln\left(\ln\left(x\right)\right)}$

$=\dfrac{1}{x{\cdot}\ln\left(x\right){\cdot}\ln\left(\ln\left(x\right)\right)}$