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Pochodna funkcji 2lnsinxcosx

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

$f\left(x\right) =$

$2{\cdot}\cos\left(x\right){\cdot}\ln\left(\sin\left(x\right)\right)$

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

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

$=\class{steps-node}{\cssId{steps-node-2}{2{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right){\cdot}\ln\left(\sin\left(x\right)\right)\right)}}}}$

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

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

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

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

Wynik alternatywny:

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