gdzie jesteś: Start / Pochodne / Pochodne funkcji złożonych / Pochodna funkcji x^2*sin(1/x)

Pochodna funkcji x^2*sin(1/x)

$f\left(x\right) =$	$\sin\left(\dfrac{1}{x}\right){\cdot}{x}^{2}$ Note: Your input has been rewritten/simplified.
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(\dfrac{1}{x}\right){\cdot}{x}^{2}\right)}}$ $=\class{steps-node}{\cssId{steps-node-3}{\class{steps-node}{\cssId{steps-node-2}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(\dfrac{1}{x}\right)\right)}}{\cdot}{x}^{2}}}+\class{steps-node}{\cssId{steps-node-5}{\sin\left(\dfrac{1}{x}\right){\cdot}\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}}}$ $=\class{steps-node}{\cssId{steps-node-6}{\cos\left(\dfrac{1}{x}\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-7}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{1}{x}\right)}}{\cdot}{x}^{2}+\class{steps-node}{\cssId{steps-node-8}{2}}\class{steps-node}{\cssId{steps-node-9}{x}}{\cdot}\sin\left(\dfrac{1}{x}\right)$ $=\dfrac{\class{steps-node}{\cssId{steps-node-12}{-\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}}{\class{steps-node}{\cssId{steps-node-10}{{x}^{2}}}}{\cdot}\cos\left(\dfrac{1}{x}\right){\cdot}{x}^{2}+2{\cdot}\sin\left(\dfrac{1}{x}\right){\cdot}x$ $=2{\cdot}\sin\left(\dfrac{1}{x}\right){\cdot}x-\class{steps-node}{\cssId{steps-node-13}{1}}{\cdot}\cos\left(\dfrac{1}{x}\right)$ $=2{\cdot}\sin\left(\dfrac{1}{x}\right){\cdot}x-\cos\left(\dfrac{1}{x}\right)$

$f\left(x\right) =$

$\sin\left(\dfrac{1}{x}\right){\cdot}{x}^{2}$

Note: Your input has been rewritten/simplified.

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

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\sin\left(\dfrac{1}{x}\right){\cdot}{x}^{2}\right)}}$

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-12}{-\class{steps-node}{\cssId{steps-node-11}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}}{\class{steps-node}{\cssId{steps-node-10}{{x}^{2}}}}{\cdot}\cos\left(\dfrac{1}{x}\right){\cdot}{x}^{2}+2{\cdot}\sin\left(\dfrac{1}{x}\right){\cdot}x$

$=2{\cdot}\sin\left(\dfrac{1}{x}\right){\cdot}x-\class{steps-node}{\cssId{steps-node-13}{1}}{\cdot}\cos\left(\dfrac{1}{x}\right)$

$=2{\cdot}\sin\left(\dfrac{1}{x}\right){\cdot}x-\cos\left(\dfrac{1}{x}\right)$