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Pochodna funkcji x/sinx

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

$f\left(x\right) =$

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

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

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-7}{1}}{\cdot}\sin\left(x\right)-\class{steps-node}{\cssId{steps-node-8}{\cos\left(x\right)}}{\cdot}x}{{\left(\sin\left(x\right)\right)}^{2}}$

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

Wynik alternatywny:

$=\dfrac{1}{\sin\left(x\right)}-\dfrac{x{\cdot}\cos\left(x\right)}{{\left(\sin\left(x\right)\right)}^{2}}$