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Pochodna funkcji arcsin(1/x)

$f\left(x\right) =$	$\arcsin\left(\dfrac{1}{x}\right)$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\arcsin\left(\dfrac{1}{x}\right)\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{\dfrac{1}{\sqrt{1-{\left(\dfrac{1}{x}\right)}^{2}}}}}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{1}{x}\right)}}$ $=\dfrac{\dfrac{\class{steps-node}{\cssId{steps-node-6}{-\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}}{\class{steps-node}{\cssId{steps-node-4}{{x}^{2}}}}}{\sqrt{1-\dfrac{1}{{x}^{2}}}}$ $=\dfrac{-\class{steps-node}{\cssId{steps-node-7}{1}}}{\sqrt{1-\dfrac{1}{{x}^{2}}}{\cdot}{x}^{2}}$

$f\left(x\right) =$

$\arcsin\left(\dfrac{1}{x}\right)$

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

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

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

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

$=\dfrac{-\class{steps-node}{\cssId{steps-node-7}{1}}}{\sqrt{1-\dfrac{1}{{x}^{2}}}{\cdot}{x}^{2}}$