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Pochodna funkcji cosx^2/x

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

$f\left(x\right) =$

$\dfrac{{\left(\cos\left(x\right)\right)}^{2}}{x}$

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

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-7}{2}}{\cdot}\class{steps-node}{\cssId{steps-node-8}{\cos\left(x\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}x-\class{steps-node}{\cssId{steps-node-10}{1}}{\cdot}{\left(\cos\left(x\right)\right)}^{2}}{{x}^{2}}$

$=\dfrac{2{\cdot}\class{steps-node}{\cssId{steps-node-11}{\left(-\sin\left(x\right)\right)}}{\cdot}x{\cdot}\cos\left(x\right)-{\left(\cos\left(x\right)\right)}^{2}}{{x}^{2}}$

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

Wynik alternatywny:

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