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Pochodna funkcji (x^3+1)/(x+2)

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

$f\left(x\right) =$

$\dfrac{{x}^{3}+1}{x+2}$

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

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

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

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

$=\dfrac{-{x}^{3}+\class{steps-node}{\cssId{steps-node-9}{3}}\class{steps-node}{\cssId{steps-node-10}{{x}^{2}}}{\cdot}\left(x+2\right)-1}{{\left(x+2\right)}^{2}}$

Uproszczony wynik:

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