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Pochodna funkcji cos(t/x)

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

$f\left(t, x\right) =$

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

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

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

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

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

$=-t{\cdot}\dfrac{\class{steps-node}{\cssId{steps-node-8}{-\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}}{\class{steps-node}{\cssId{steps-node-6}{{x}^{2}}}}{\cdot}\sin\left(\dfrac{t}{x}\right)$

$=\dfrac{t{\cdot}\class{steps-node}{\cssId{steps-node-9}{1}}{\cdot}\sin\left(\dfrac{t}{x}\right)}{{x}^{2}}$

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