gdzie jesteś: Start / Pochodne / Pochodne funkcji trygonometrycznych / Pochodna funkcji lna/lnx

Pochodna funkcji lna/lnx

$f\left(a, x\right) =$	$\dfrac{\ln\left(a\right)}{\ln\left(x\right)}$
$\dfrac{\mathrm{d}\left(f\left(a, x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{\ln\left(a\right)}{\ln\left(x\right)}\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{\ln\left(a\right){\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{1}{\ln\left(x\right)}\right)}}}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-6}{-\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}}{\class{steps-node}{\cssId{steps-node-4}{{\left(\ln\left(x\right)\right)}^{2}}}}{\cdot}\ln\left(a\right)$ $=\dfrac{-\class{steps-node}{\cssId{steps-node-7}{\dfrac{1}{x}}}{\cdot}\ln\left(a\right)}{{\left(\ln\left(x\right)\right)}^{2}}$ $=\dfrac{-\ln\left(a\right)}{x{\cdot}{\left(\ln\left(x\right)\right)}^{2}}$

$f\left(a, x\right) =$

$\dfrac{\ln\left(a\right)}{\ln\left(x\right)}$

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

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

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

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

$=\dfrac{-\class{steps-node}{\cssId{steps-node-7}{\dfrac{1}{x}}}{\cdot}\ln\left(a\right)}{{\left(\ln\left(x\right)\right)}^{2}}$

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