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

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

$f\left(x\right) =$

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

Note: Your input has been rewritten/simplified.

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

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

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

$=\dfrac{\dfrac{\class{steps-node}{\cssId{steps-node-6}{\left(\sqrt{{x}^{2}+1}+1\right){\cdot}\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}-\class{steps-node}{\cssId{steps-node-8}{\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sqrt{{x}^{2}+1}+1\right)}}{\cdot}x}}}{\class{steps-node}{\cssId{steps-node-4}{{\left(\sqrt{{x}^{2}+1}+1\right)}^{2}}}}}{\dfrac{{x}^{2}}{{\left(\sqrt{{x}^{2}+1}+1\right)}^{2}}+1}$

$=\dfrac{\class{steps-node}{\cssId{steps-node-9}{1}}{\cdot}\left(\sqrt{{x}^{2}+1}+1\right)-\class{steps-node}{\cssId{steps-node-10}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sqrt{{x}^{2}+1}\right)}}{\cdot}x}{{\left(\sqrt{{x}^{2}+1}+1\right)}^{2}{\cdot}\left(\dfrac{{x}^{2}}{{\left(\sqrt{{x}^{2}+1}+1\right)}^{2}}+1\right)}$

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

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

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

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

Wynik alternatywny:

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