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Pochodna funkcji 1/arcsin(x)

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

$f\left(x\right) =$

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

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

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

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

$=\dfrac{-\class{steps-node}{\cssId{steps-node-5}{\dfrac{1}{\sqrt{1-{x}^{2}}}}}}{{\left(\arcsin\left(x\right)\right)}^{2}}$

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