gdzie jesteś: Start / Pochodne / Pochodne funkcji złożonych / Pochodna funkcji x*cos(3x+1)

Pochodna funkcji x*cos(3x+1)

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

$f\left(x\right) =$

$x{\cdot}\cos\left(3x+1\right)$

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

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x{\cdot}\cos\left(3x+1\right)\right)}}$

$=\class{steps-node}{\cssId{steps-node-3}{\class{steps-node}{\cssId{steps-node-2}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\cos\left(3x+1\right)}}+\class{steps-node}{\cssId{steps-node-5}{x{\cdot}\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\cos\left(3x+1\right)\right)}}}}$

$=\class{steps-node}{\cssId{steps-node-6}{1}}{\cdot}\cos\left(3x+1\right)+\class{steps-node}{\cssId{steps-node-7}{-\sin\left(3x+1\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(3x+1\right)}}{\cdot}x$

$=\cos\left(3x+1\right)-\class{steps-node}{\cssId{steps-node-9}{3}}x{\cdot}\sin\left(3x+1\right)$