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Pochodna funkcji arctan((1-2cosx)/(pi-3x))

$f\left(x\right) =$	$\arctan\left(\dfrac{1-2{\cdot}\cos\left(x\right)}{{\pi}-3x}\right)$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\arctan\left(\dfrac{1-2{\cdot}\cos\left(x\right)}{{\pi}-3x}\right)\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{\dfrac{1}{{\left(\dfrac{1-2{\cdot}\cos\left(x\right)}{{\pi}-3x}\right)}^{2}+1}}}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{1-2{\cdot}\cos\left(x\right)}{{\pi}-3x}\right)}}$ $=\dfrac{\dfrac{\class{steps-node}{\cssId{steps-node-6}{\left({\pi}-3x\right){\cdot}\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(1-2{\cdot}\cos\left(x\right)\right)}}}}-\class{steps-node}{\cssId{steps-node-8}{\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\pi}-3x\right)}}{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)}}}{\class{steps-node}{\cssId{steps-node-4}{{\left({\pi}-3x\right)}^{2}}}}}{\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-9}{-2{\cdot}\class{steps-node}{\cssId{steps-node-10}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}}}{\cdot}\left({\pi}-3x\right)-\class{steps-node}{\cssId{steps-node-11}{-3}}{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)}{{\left({\pi}-3x\right)}^{2}{\cdot}\left(\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1\right)}$ $=\dfrac{3{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)-2{\cdot}\class{steps-node}{\cssId{steps-node-12}{\left(-\sin\left(x\right)\right)}}{\cdot}\left({\pi}-3x\right)}{{\left({\pi}-3x\right)}^{2}{\cdot}\left(\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1\right)}$ $=\dfrac{2{\cdot}\left({\pi}-3x\right){\cdot}\sin\left(x\right)+3{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)}{{\left({\pi}-3x\right)}^{2}{\cdot}\left(\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1\right)}$ Wynik alternatywny: $=\dfrac{\dfrac{2{\cdot}\sin\left(x\right)}{{\pi}-3x}+\dfrac{3{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)}{{\left({\pi}-3x\right)}^{2}}}{\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1}$

$f\left(x\right) =$

$\arctan\left(\dfrac{1-2{\cdot}\cos\left(x\right)}{{\pi}-3x}\right)$

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

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

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

$=\dfrac{\dfrac{\class{steps-node}{\cssId{steps-node-6}{\left({\pi}-3x\right){\cdot}\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(1-2{\cdot}\cos\left(x\right)\right)}}}}-\class{steps-node}{\cssId{steps-node-8}{\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\pi}-3x\right)}}{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)}}}{\class{steps-node}{\cssId{steps-node-4}{{\left({\pi}-3x\right)}^{2}}}}}{\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1}$

$=\dfrac{\class{steps-node}{\cssId{steps-node-9}{-2{\cdot}\class{steps-node}{\cssId{steps-node-10}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}}}{\cdot}\left({\pi}-3x\right)-\class{steps-node}{\cssId{steps-node-11}{-3}}{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)}{{\left({\pi}-3x\right)}^{2}{\cdot}\left(\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1\right)}$

$=\dfrac{3{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)-2{\cdot}\class{steps-node}{\cssId{steps-node-12}{\left(-\sin\left(x\right)\right)}}{\cdot}\left({\pi}-3x\right)}{{\left({\pi}-3x\right)}^{2}{\cdot}\left(\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1\right)}$

$=\dfrac{2{\cdot}\left({\pi}-3x\right){\cdot}\sin\left(x\right)+3{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)}{{\left({\pi}-3x\right)}^{2}{\cdot}\left(\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1\right)}$

Wynik alternatywny:

$=\dfrac{\dfrac{2{\cdot}\sin\left(x\right)}{{\pi}-3x}+\dfrac{3{\cdot}\left(1-2{\cdot}\cos\left(x\right)\right)}{{\left({\pi}-3x\right)}^{2}}}{\dfrac{{\left(1-2{\cdot}\cos\left(x\right)\right)}^{2}}{{\left({\pi}-3x\right)}^{2}}+1}$