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Pochodna funkcji (x^4-15x^2+6x)/(x^2-5)^2

$f\left(x\right) =$	$\dfrac{{x}^{4}-15{x}^{2}+6x}{{\left({x}^{2}-5\right)}^{2}}$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{{x}^{4}-15{x}^{2}+6x}{{\left({x}^{2}-5\right)}^{2}}\right)}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-4}{{\left({x}^{2}-5\right)}^{2}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{4}-15{x}^{2}+6x\right)}}}}-\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left({x}^{2}-5\right)}^{2}\right)}}{\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}}}{\class{steps-node}{\cssId{steps-node-2}{{\left({\left({x}^{2}-5\right)}^{2}\right)}^{2}}}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-7}{\left(\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{4}\right)}}-15{\cdot}\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}+6\right)}}{\cdot}{\left({x}^{2}-5\right)}^{2}-\class{steps-node}{\cssId{steps-node-10}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\left({x}^{2}-5\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-12}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}-5\right)}}{\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{4}}$ $=\dfrac{\left(-15{\cdot}\class{steps-node}{\cssId{steps-node-15}{2}}\class{steps-node}{\cssId{steps-node-16}{x}}+\class{steps-node}{\cssId{steps-node-13}{4}}\class{steps-node}{\cssId{steps-node-14}{{x}^{3}}}+6\right){\cdot}{\left({x}^{2}-5\right)}^{2}-2{\cdot}\class{steps-node}{\cssId{steps-node-17}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}{\cdot}\left({x}^{2}-5\right){\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{4}}$ $=\dfrac{{\left({x}^{2}-5\right)}^{2}{\cdot}\left(4{x}^{3}-30x+6\right)-2{\cdot}\class{steps-node}{\cssId{steps-node-18}{2}}\class{steps-node}{\cssId{steps-node-19}{x}}{\cdot}\left({x}^{2}-5\right){\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{4}}$ $=\dfrac{{\left({x}^{2}-5\right)}^{2}{\cdot}\left(4{x}^{3}-30x+6\right)-4x{\cdot}\left({x}^{2}-5\right){\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{4}}$ Uproszczony wynik: $=\dfrac{4{x}^{3}-30x+6}{{\left({x}^{2}-5\right)}^{2}}-\dfrac{4x{\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{3}}$

$f\left(x\right) =$

$\dfrac{{x}^{4}-15{x}^{2}+6x}{{\left({x}^{2}-5\right)}^{2}}$

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

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{{x}^{4}-15{x}^{2}+6x}{{\left({x}^{2}-5\right)}^{2}}\right)}}$

$=\dfrac{\class{steps-node}{\cssId{steps-node-4}{{\left({x}^{2}-5\right)}^{2}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{4}-15{x}^{2}+6x\right)}}}}-\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left({x}^{2}-5\right)}^{2}\right)}}{\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}}}{\class{steps-node}{\cssId{steps-node-2}{{\left({\left({x}^{2}-5\right)}^{2}\right)}^{2}}}}$

$=\dfrac{\class{steps-node}{\cssId{steps-node-7}{\left(\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{4}\right)}}-15{\cdot}\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}+6\right)}}{\cdot}{\left({x}^{2}-5\right)}^{2}-\class{steps-node}{\cssId{steps-node-10}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\left({x}^{2}-5\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-12}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}-5\right)}}{\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{4}}$

$=\dfrac{\left(-15{\cdot}\class{steps-node}{\cssId{steps-node-15}{2}}\class{steps-node}{\cssId{steps-node-16}{x}}+\class{steps-node}{\cssId{steps-node-13}{4}}\class{steps-node}{\cssId{steps-node-14}{{x}^{3}}}+6\right){\cdot}{\left({x}^{2}-5\right)}^{2}-2{\cdot}\class{steps-node}{\cssId{steps-node-17}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}{\cdot}\left({x}^{2}-5\right){\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{4}}$

$=\dfrac{{\left({x}^{2}-5\right)}^{2}{\cdot}\left(4{x}^{3}-30x+6\right)-2{\cdot}\class{steps-node}{\cssId{steps-node-18}{2}}\class{steps-node}{\cssId{steps-node-19}{x}}{\cdot}\left({x}^{2}-5\right){\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{4}}$

$=\dfrac{{\left({x}^{2}-5\right)}^{2}{\cdot}\left(4{x}^{3}-30x+6\right)-4x{\cdot}\left({x}^{2}-5\right){\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{4}}$

Uproszczony wynik:

$=\dfrac{4{x}^{3}-30x+6}{{\left({x}^{2}-5\right)}^{2}}-\dfrac{4x{\cdot}\left({x}^{4}-15{x}^{2}+6x\right)}{{\left({x}^{2}-5\right)}^{3}}$