gdzie jesteś: Start / Pochodne / Pochodne funkcji trygonometrycznych / Pochodna funkcji ln(x-1)/(x^2+1)

Pochodna funkcji ln(x-1)/(x^2+1)

$f\left(x\right) =$	$\dfrac{\ln\left(x-1\right)}{{x}^{2}+1}$
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{\ln\left(x-1\right)}{{x}^{2}+1}\right)}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-4}{\left({x}^{2}+1\right){\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x-1\right)\right)}}}}-\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}+1\right)}}{\cdot}\ln\left(x-1\right)}}}{\class{steps-node}{\cssId{steps-node-2}{{\left({x}^{2}+1\right)}^{2}}}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-7}{\dfrac{1}{x-1}}}{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}{\cdot}\left({x}^{2}+1\right)-\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}\right)}}{\cdot}\ln\left(x-1\right)}{{\left({x}^{2}+1\right)}^{2}}$ $=\dfrac{\dfrac{\class{steps-node}{\cssId{steps-node-10}{1}}{\cdot}\left({x}^{2}+1\right)}{x-1}-\class{steps-node}{\cssId{steps-node-11}{2}}\class{steps-node}{\cssId{steps-node-12}{x}}{\cdot}\ln\left(x-1\right)}{{\left({x}^{2}+1\right)}^{2}}$ $=\dfrac{\dfrac{{x}^{2}+1}{x-1}-2x{\cdot}\ln\left(x-1\right)}{{\left({x}^{2}+1\right)}^{2}}$ Wynik alternatywny: $=\dfrac{1}{\left(x-1\right){\cdot}\left({x}^{2}+1\right)}-\dfrac{2x{\cdot}\ln\left(x-1\right)}{{\left({x}^{2}+1\right)}^{2}}$

$f\left(x\right) =$

$\dfrac{\ln\left(x-1\right)}{{x}^{2}+1}$

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-4}{\left({x}^{2}+1\right){\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x-1\right)\right)}}}}-\class{steps-node}{\cssId{steps-node-6}{\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{2}+1\right)}}{\cdot}\ln\left(x-1\right)}}}{\class{steps-node}{\cssId{steps-node-2}{{\left({x}^{2}+1\right)}^{2}}}}$

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

$=\dfrac{\dfrac{\class{steps-node}{\cssId{steps-node-10}{1}}{\cdot}\left({x}^{2}+1\right)}{x-1}-\class{steps-node}{\cssId{steps-node-11}{2}}\class{steps-node}{\cssId{steps-node-12}{x}}{\cdot}\ln\left(x-1\right)}{{\left({x}^{2}+1\right)}^{2}}$

$=\dfrac{\dfrac{{x}^{2}+1}{x-1}-2x{\cdot}\ln\left(x-1\right)}{{\left({x}^{2}+1\right)}^{2}}$

Wynik alternatywny:

$=\dfrac{1}{\left(x-1\right){\cdot}\left({x}^{2}+1\right)}-\dfrac{2x{\cdot}\ln\left(x-1\right)}{{\left({x}^{2}+1\right)}^{2}}$