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Pochodna funkcji x^(5*x)

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

$f\left(x\right) =$

${x}^{5x}$

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

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{5x}\right)}}$

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

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

$=5{\cdot}\left(\class{steps-node}{\cssId{steps-node-7}{\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}+\class{steps-node}{\cssId{steps-node-9}{x{\cdot}\class{steps-node}{\cssId{steps-node-8}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}\right){\cdot}{x}^{5x}$

$=5{x}^{5x}{\cdot}\left(\class{steps-node}{\cssId{steps-node-10}{1}}{\cdot}\ln\left(x\right)+\class{steps-node}{\cssId{steps-node-11}{\dfrac{1}{x}}}{\cdot}x\right)$

$=5{x}^{5x}{\cdot}\left(\ln\left(x\right)+1\right)$

Wynik alternatywny:

$={x}^{5x}{\cdot}\left(5{\cdot}\ln\left(x\right)+5\right)$