gdzie jesteś: Start / Pochodne / Pochodne funkcji trygonometrycznych / Pochodna funkcji 64*(30*((cosx)^8)*((sinx)^4)-84*((cosx)^6)*((sinx)^6)+30*((cosx)^8)*((sinx)^4))

Pochodna funkcji 64(30((cosx)^8)((sinx)^4)-84((cosx)^6)((sinx)^6)+30((cosx)^8)*((sinx)^4))

$f\left(x\right) =$	$64{\cdot}\left(60{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}-84{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)$ Note: Your input has been rewritten/simplified.
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(64{\cdot}\left(60{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}-84{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{64{\cdot}\left(60{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}\right)}}-84{\cdot}\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)}}\right)}}$ $=64{\cdot}\left(60{\cdot}\left(\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{8}\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{4}}}+\class{steps-node}{\cssId{steps-node-8}{{\left(\cos\left(x\right)\right)}^{8}{\cdot}\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\sin\left(x\right)\right)}^{4}\right)}}}}\right)-84{\cdot}\left(\class{steps-node}{\cssId{steps-node-10}{\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{6}\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{6}}}+\class{steps-node}{\cssId{steps-node-12}{{\left(\cos\left(x\right)\right)}^{6}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\sin\left(x\right)\right)}^{6}\right)}}}}\right)\right)$ $=64{\cdot}\left(60{\cdot}\left(\class{steps-node}{\cssId{steps-node-13}{8}}{\cdot}\class{steps-node}{\cssId{steps-node-14}{{\left(\cos\left(x\right)\right)}^{7}}}{\cdot}\class{steps-node}{\cssId{steps-node-15}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{4}+\class{steps-node}{\cssId{steps-node-16}{4}}{\cdot}\class{steps-node}{\cssId{steps-node-17}{{\left(\sin\left(x\right)\right)}^{3}}}{\cdot}\class{steps-node}{\cssId{steps-node-18}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{8}\right)-84{\cdot}\left(\class{steps-node}{\cssId{steps-node-19}{6}}{\cdot}\class{steps-node}{\cssId{steps-node-20}{{\left(\cos\left(x\right)\right)}^{5}}}{\cdot}\class{steps-node}{\cssId{steps-node-21}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{6}+\class{steps-node}{\cssId{steps-node-22}{6}}{\cdot}\class{steps-node}{\cssId{steps-node-23}{{\left(\sin\left(x\right)\right)}^{5}}}{\cdot}\class{steps-node}{\cssId{steps-node-24}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{6}\right)\right)$ $=64{\cdot}\left(60{\cdot}\left(8{\cdot}\class{steps-node}{\cssId{steps-node-25}{\left(-\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{4}+4{\cdot}\class{steps-node}{\cssId{steps-node-26}{\cos\left(x\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{3}\right)-84{\cdot}\left(6{\cdot}\class{steps-node}{\cssId{steps-node-27}{\left(-\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{6}+6{\cdot}\class{steps-node}{\cssId{steps-node-28}{\cos\left(x\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{5}\right)\right)$ $=64{\cdot}\left(60{\cdot}\left(4{\cdot}{\left(\cos\left(x\right)\right)}^{9}{\cdot}{\left(\sin\left(x\right)\right)}^{3}-8{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}\right)-84{\cdot}\left(6{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}-6{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}\right)\right)$ Uproszczony wynik: $=64{\cdot}\left(504{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}-984{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}+240{\cdot}{\left(\cos\left(x\right)\right)}^{9}{\cdot}{\left(\sin\left(x\right)\right)}^{3}\right)$

$f\left(x\right) =$

$64{\cdot}\left(60{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}-84{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)$

Note: Your input has been rewritten/simplified.

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

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(64{\cdot}\left(60{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}-84{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)\right)}}$

$=\class{steps-node}{\cssId{steps-node-2}{64{\cdot}\left(60{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{4}\right)}}-84{\cdot}\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{6}\right)}}\right)}}$

$=64{\cdot}\left(60{\cdot}\left(\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{8}\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{4}}}+\class{steps-node}{\cssId{steps-node-8}{{\left(\cos\left(x\right)\right)}^{8}{\cdot}\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\sin\left(x\right)\right)}^{4}\right)}}}}\right)-84{\cdot}\left(\class{steps-node}{\cssId{steps-node-10}{\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\cos\left(x\right)\right)}^{6}\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{6}}}+\class{steps-node}{\cssId{steps-node-12}{{\left(\cos\left(x\right)\right)}^{6}{\cdot}\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({\left(\sin\left(x\right)\right)}^{6}\right)}}}}\right)\right)$

$=64{\cdot}\left(60{\cdot}\left(\class{steps-node}{\cssId{steps-node-13}{8}}{\cdot}\class{steps-node}{\cssId{steps-node-14}{{\left(\cos\left(x\right)\right)}^{7}}}{\cdot}\class{steps-node}{\cssId{steps-node-15}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{4}+\class{steps-node}{\cssId{steps-node-16}{4}}{\cdot}\class{steps-node}{\cssId{steps-node-17}{{\left(\sin\left(x\right)\right)}^{3}}}{\cdot}\class{steps-node}{\cssId{steps-node-18}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{8}\right)-84{\cdot}\left(\class{steps-node}{\cssId{steps-node-19}{6}}{\cdot}\class{steps-node}{\cssId{steps-node-20}{{\left(\cos\left(x\right)\right)}^{5}}}{\cdot}\class{steps-node}{\cssId{steps-node-21}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(x\right)\right)}}{\cdot}{\left(\sin\left(x\right)\right)}^{6}+\class{steps-node}{\cssId{steps-node-22}{6}}{\cdot}\class{steps-node}{\cssId{steps-node-23}{{\left(\sin\left(x\right)\right)}^{5}}}{\cdot}\class{steps-node}{\cssId{steps-node-24}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{6}\right)\right)$

$=64{\cdot}\left(60{\cdot}\left(8{\cdot}\class{steps-node}{\cssId{steps-node-25}{\left(-\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{4}+4{\cdot}\class{steps-node}{\cssId{steps-node-26}{\cos\left(x\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{8}{\cdot}{\left(\sin\left(x\right)\right)}^{3}\right)-84{\cdot}\left(6{\cdot}\class{steps-node}{\cssId{steps-node-27}{\left(-\sin\left(x\right)\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{6}+6{\cdot}\class{steps-node}{\cssId{steps-node-28}{\cos\left(x\right)}}{\cdot}{\left(\cos\left(x\right)\right)}^{6}{\cdot}{\left(\sin\left(x\right)\right)}^{5}\right)\right)$

$=64{\cdot}\left(60{\cdot}\left(4{\cdot}{\left(\cos\left(x\right)\right)}^{9}{\cdot}{\left(\sin\left(x\right)\right)}^{3}-8{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}\right)-84{\cdot}\left(6{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}-6{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}\right)\right)$

Uproszczony wynik:

$=64{\cdot}\left(504{\cdot}{\left(\cos\left(x\right)\right)}^{5}{\cdot}{\left(\sin\left(x\right)\right)}^{7}-984{\cdot}{\left(\cos\left(x\right)\right)}^{7}{\cdot}{\left(\sin\left(x\right)\right)}^{5}+240{\cdot}{\left(\cos\left(x\right)\right)}^{9}{\cdot}{\left(\sin\left(x\right)\right)}^{3}\right)$