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Pochodna funkcji x*sin(1/x)

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

$f\left(x\right) =$

$\sin\left(\dfrac{1}{x}\right){\cdot}x$

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

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

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

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

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

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

$=\sin\left(\dfrac{1}{x}\right)-\dfrac{\cos\left(\dfrac{1}{x}\right)}{x}$