gdzie jesteś: Start / Pochodne / Pochodne funkcji trygonometrycznych / Pochodna funkcji 64(6*((cosx)^7)*((sinx)^5)-6*((cosx)^5)*((sinx)^7))

Pochodna funkcji 64(6((cosx)^7)((sinx)^5)-6((cosx)^5)((sinx)^7))

$f\left(x\right) =$	$64{\cdot}\left(6{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}-6{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}\right)$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(64{\cdot}\left(6{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}-6{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}\right)\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{64{\cdot}\left(6{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}\right)}}-6{\cdot}\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}\right)}}\right)}}$ $=64{\cdot}\left(6{\cdot}\left(\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{7}\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{5}}}+\class{steps-node}{\cssId{steps-node-8}{{\left(\cos\left(x\right)\right)}^{7}{\cdot}\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\sin\left(x\right)\right)}^{5}\right)}}}}\right)-6{\cdot}\left(\class{steps-node}{\cssId{steps-node-10}{\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{5}\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{7}}}+\class{steps-node}{\cssId{steps-node-12}{{\left(\cos\left(x\right)\right)}^{5}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\sin\left(x\right)\right)}^{7}\right)}}}}\right)\right)$ $=64{\cdot}\left(6{\cdot}\left(\class{steps-node}{\cssId{steps-node-13}{7}}{\cdot}\class{steps-node}{\cssId{steps-node-14}{{\left(\cos\left(x\right)\right)}^{6}}}{\cdot}\class{steps-node}{\cssId{steps-node-15}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{5}+\class{steps-node}{\cssId{steps-node-16}{5}}{\cdot}\class{steps-node}{\cssId{steps-node-17}{{\left(\sin\left(x\right)\right)}^{4}}}{\cdot}\class{steps-node}{\cssId{steps-node-18}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{7}\right)-6{\cdot}\left(\class{steps-node}{\cssId{steps-node-19}{5}}{\cdot}\class{steps-node}{\cssId{steps-node-20}{{\left(\cos\left(x\right)\right)}^{4}}}{\cdot}\class{steps-node}{\cssId{steps-node-21}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{7}+\class{steps-node}{\cssId{steps-node-22}{7}}{\cdot}\class{steps-node}{\cssId{steps-node-23}{{\left(\sin\left(x\right)\right)}^{6}}}{\cdot}\class{steps-node}{\cssId{steps-node-24}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{5}\right)\right)$ $=64{\cdot}\left(6{\cdot}\left(7{\cdot}\class{steps-node}{\cssId{steps-node-25}{\left(-\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{5}+5{\cdot}\class{steps-node}{\cssId{steps-node-26}{\cos\left(x\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{4}\right)-6{\cdot}\left(5{\cdot}\class{steps-node}{\cssId{steps-node-27}{\left(-\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{4}{\cdot}{\left(\sin\left(x\right)\right)}^{7}+7{\cdot}\class{steps-node}{\cssId{steps-node-28}{\cos\left(x\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)\right)$ $=64{\cdot}\left(6{\cdot}\left(5{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}-7{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)-6{\cdot}\left(7{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}-5{\cdot}{\left(\cos\left(x\right)\right)}^{4}{\cdot}{\left(\sin\left(x\right)\right)}^{8}\right)\right)$ Uproszczony wynik: $=64{\cdot}\left(30{\cdot}{\left(\cos\left(x\right)\right)}^{4}{\cdot}{\left(\sin\left(x\right)\right)}^{8}-84{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}+30{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}\right)$

$f\left(x\right) =$

$64{\cdot}\left(6{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}-6{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}\right)$

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

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(64{\cdot}\left(6{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}-6{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}\right)\right)}}$

$=\class{steps-node}{\cssId{steps-node-2}{64{\cdot}\left(6{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}\right)}}-6{\cdot}\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}\right)}}\right)}}$

$=64{\cdot}\left(6{\cdot}\left(\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{7}\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{5}}}+\class{steps-node}{\cssId{steps-node-8}{{\left(\cos\left(x\right)\right)}^{7}{\cdot}\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\sin\left(x\right)\right)}^{5}\right)}}}}\right)-6{\cdot}\left(\class{steps-node}{\cssId{steps-node-10}{\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{5}\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{7}}}+\class{steps-node}{\cssId{steps-node-12}{{\left(\cos\left(x\right)\right)}^{5}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\sin\left(x\right)\right)}^{7}\right)}}}}\right)\right)$

$=64{\cdot}\left(6{\cdot}\left(\class{steps-node}{\cssId{steps-node-13}{7}}{\cdot}\class{steps-node}{\cssId{steps-node-14}{{\left(\cos\left(x\right)\right)}^{6}}}{\cdot}\class{steps-node}{\cssId{steps-node-15}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{5}+\class{steps-node}{\cssId{steps-node-16}{5}}{\cdot}\class{steps-node}{\cssId{steps-node-17}{{\left(\sin\left(x\right)\right)}^{4}}}{\cdot}\class{steps-node}{\cssId{steps-node-18}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{7}\right)-6{\cdot}\left(\class{steps-node}{\cssId{steps-node-19}{5}}{\cdot}\class{steps-node}{\cssId{steps-node-20}{{\left(\cos\left(x\right)\right)}^{4}}}{\cdot}\class{steps-node}{\cssId{steps-node-21}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{7}+\class{steps-node}{\cssId{steps-node-22}{7}}{\cdot}\class{steps-node}{\cssId{steps-node-23}{{\left(\sin\left(x\right)\right)}^{6}}}{\cdot}\class{steps-node}{\cssId{steps-node-24}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{5}\right)\right)$

$=64{\cdot}\left(6{\cdot}\left(7{\cdot}\class{steps-node}{\cssId{steps-node-25}{\left(-\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{5}+5{\cdot}\class{steps-node}{\cssId{steps-node-26}{\cos\left(x\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{4}\right)-6{\cdot}\left(5{\cdot}\class{steps-node}{\cssId{steps-node-27}{\left(-\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{4}{\cdot}{\left(\sin\left(x\right)\right)}^{7}+7{\cdot}\class{steps-node}{\cssId{steps-node-28}{\cos\left(x\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)\right)$

$=64{\cdot}\left(6{\cdot}\left(5{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}-7{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)-6{\cdot}\left(7{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}-5{\cdot}{\left(\cos\left(x\right)\right)}^{4}{\cdot}{\left(\sin\left(x\right)\right)}^{8}\right)\right)$

Uproszczony wynik:

$=64{\cdot}\left(30{\cdot}{\left(\cos\left(x\right)\right)}^{4}{\cdot}{\left(\sin\left(x\right)\right)}^{8}-84{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}+30{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}\right)$