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Pochodna funkcji arctan(x(-sqrt(x*x+1)))

$f\left(x\right) =$	$-\arctan\left(x{\cdot}\sqrt{{x}^{2}+1}\right)$ Note: Your input has been rewritten/simplified.
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(-\arctan\left(x{\cdot}\sqrt{{x}^{2}+1}\right)\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{-\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\arctan\left(x{\cdot}\sqrt{{x}^{2}+1}\right)\right)}}}}$ $=-\left(\class{steps-node}{\cssId{steps-node-4}{\dfrac{1}{{\left(x{\cdot}\sqrt{{x}^{2}+1}\right)}^{2}+1}}}{\cdot}\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x{\cdot}\sqrt{{x}^{2}+1}\right)}}\right)$ $=\dfrac{-\left(\class{steps-node}{\cssId{steps-node-7}{\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\sqrt{{x}^{2}+1}}}+\class{steps-node}{\cssId{steps-node-9}{x{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sqrt{{x}^{2}+1}\right)}}}}\right)}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$ $=\dfrac{-\class{steps-node}{\cssId{steps-node-10}{1}}{\cdot}\sqrt{{x}^{2}+1}-\class{steps-node}{\cssId{steps-node-11}{\dfrac{1}{2{\cdot}\sqrt{{x}^{2}+1}}}}{\cdot}\class{steps-node}{\cssId{steps-node-12}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}+1\right)}}{\cdot}x}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$ $=\dfrac{-\sqrt{{x}^{2}+1}-\dfrac{\class{steps-node}{\cssId{steps-node-13}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}{\cdot}x}{2{\cdot}\sqrt{{x}^{2}+1}}}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$ $=\dfrac{-\sqrt{{x}^{2}+1}-\dfrac{\class{steps-node}{\cssId{steps-node-14}{2}}\class{steps-node}{\cssId{steps-node-15}{x}}{\cdot}x}{2{\cdot}\sqrt{{x}^{2}+1}}}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$ $=\dfrac{-\sqrt{{x}^{2}+1}-\dfrac{{x}^{2}}{\sqrt{{x}^{2}+1}}}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$ Uproszczony wynik: $=\dfrac{-\left(\sqrt{{x}^{2}+1}+\dfrac{{x}^{2}}{\sqrt{{x}^{2}+1}}\right)}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$

$f\left(x\right) =$

$-\arctan\left(x{\cdot}\sqrt{{x}^{2}+1}\right)$

Note: Your input has been rewritten/simplified.

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

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

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

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

$=\dfrac{-\left(\class{steps-node}{\cssId{steps-node-7}{\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\sqrt{{x}^{2}+1}}}+\class{steps-node}{\cssId{steps-node-9}{x{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sqrt{{x}^{2}+1}\right)}}}}\right)}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$

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

$=\dfrac{-\sqrt{{x}^{2}+1}-\dfrac{\class{steps-node}{\cssId{steps-node-13}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}{\cdot}x}{2{\cdot}\sqrt{{x}^{2}+1}}}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$

$=\dfrac{-\sqrt{{x}^{2}+1}-\dfrac{\class{steps-node}{\cssId{steps-node-14}{2}}\class{steps-node}{\cssId{steps-node-15}{x}}{\cdot}x}{2{\cdot}\sqrt{{x}^{2}+1}}}{{x}^{2}{\cdot}\left({x}^{2}+1\right)+1}$

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

Uproszczony wynik:

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