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Pochodna funkcji arctan(x)-arctan(1/x)

$f\left(x\right) =$	$\arctan\left(x\right)-\arctan\left(\dfrac{1}{x}\right)$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\arctan\left(x\right)-\arctan\left(\dfrac{1}{x}\right)\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\arctan\left(x\right)\right)}}-\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\arctan\left(\dfrac{1}{x}\right)\right)}}}}$ $=\class{steps-node}{\cssId{steps-node-5}{\dfrac{1}{{x}^{2}+1}}}-\class{steps-node}{\cssId{steps-node-6}{\dfrac{1}{{\left(\dfrac{1}{x}\right)}^{2}+1}}}{\cdot}\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{1}{x}\right)}}$ $=\dfrac{1}{{x}^{2}+1}-\dfrac{\dfrac{\class{steps-node}{\cssId{steps-node-10}{-\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}}{\class{steps-node}{\cssId{steps-node-8}{{x}^{2}}}}}{\dfrac{1}{{x}^{2}}+1}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-11}{1}}}{{x}^{2}+1}+\dfrac{1}{{x}^{2}+1}$ $=\dfrac{2}{{x}^{2}+1}$ Wynik alternatywny: $=\dfrac{1}{{x}^{2}+1}+\dfrac{1}{\left(\dfrac{1}{{x}^{2}}+1\right){\cdot}{x}^{2}}$

$f\left(x\right) =$

$\arctan\left(x\right)-\arctan\left(\dfrac{1}{x}\right)$

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

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

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

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

$=\dfrac{1}{{x}^{2}+1}-\dfrac{\dfrac{\class{steps-node}{\cssId{steps-node-10}{-\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}}{\class{steps-node}{\cssId{steps-node-8}{{x}^{2}}}}}{\dfrac{1}{{x}^{2}}+1}$

$=\dfrac{\class{steps-node}{\cssId{steps-node-11}{1}}}{{x}^{2}+1}+\dfrac{1}{{x}^{2}+1}$

$=\dfrac{2}{{x}^{2}+1}$

Wynik alternatywny:

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