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Pochodna funkcji (7x^3+9)/(x-5)

$f\left(x\right) =$	$\dfrac{7{x}^{3}+9}{x-5}$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{7{x}^{3}+9}{x-5}\right)}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-4}{\left(x-5\right){\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(7{x}^{3}+9\right)}}}}-\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-5\right)}}{\cdot}\left(7{x}^{3}+9\right)}}}{\class{steps-node}{\cssId{steps-node-2}{{\left(x-5\right)}^{2}}}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-7}{7{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{3}\right)}}}}{\cdot}\left(x-5\right)-\class{steps-node}{\cssId{steps-node-9}{1}}{\cdot}\left(7{x}^{3}+9\right)}{{\left(x-5\right)}^{2}}$ $=\dfrac{-7{x}^{3}+7{\cdot}\class{steps-node}{\cssId{steps-node-10}{3}}\class{steps-node}{\cssId{steps-node-11}{{x}^{2}}}{\cdot}\left(x-5\right)-9}{{\left(x-5\right)}^{2}}$ $=\dfrac{-7{x}^{3}+21{\cdot}\left(x-5\right){\cdot}{x}^{2}-9}{{\left(x-5\right)}^{2}}$ Uproszczony wynik: $=\dfrac{21{x}^{2}}{x-5}-\dfrac{7{x}^{3}+9}{{\left(x-5\right)}^{2}}$

$f\left(x\right) =$

$\dfrac{7{x}^{3}+9}{x-5}$

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

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{7{x}^{3}+9}{x-5}\right)}}$

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

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

$=\dfrac{-7{x}^{3}+7{\cdot}\class{steps-node}{\cssId{steps-node-10}{3}}\class{steps-node}{\cssId{steps-node-11}{{x}^{2}}}{\cdot}\left(x-5\right)-9}{{\left(x-5\right)}^{2}}$

$=\dfrac{-7{x}^{3}+21{\cdot}\left(x-5\right){\cdot}{x}^{2}-9}{{\left(x-5\right)}^{2}}$

Uproszczony wynik:

$=\dfrac{21{x}^{2}}{x-5}-\dfrac{7{x}^{3}+9}{{\left(x-5\right)}^{2}}$