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Pochodna funkcji sinx/cosx

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

$f\left(x\right) =$

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

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

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

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

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

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

Uproszczony wynik:

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