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Pochodna funkcji tg(2x)^x/2

$f\left(g, t, x\right) =$	$gt{x}^{x}{\cdot}{2}^{x-1}$
$\dfrac{\mathrm{d}\left(f\left(g, t, x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(gt{x}^{x}{\cdot}{2}^{x-1}\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{gt{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{x}{\cdot}{2}^{x-1}\right)}}}}$ $=gt{\cdot}\left(\class{steps-node}{\cssId{steps-node-5}{\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{x}\right)}}{\cdot}{2}^{x-1}}}+\class{steps-node}{\cssId{steps-node-7}{{x}^{x}{\cdot}\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({2}^{x-1}\right)}}}}\right)$ $=gt{\cdot}\left(\class{steps-node}{\cssId{steps-node-8}{{x}^{x}}}{\cdot}\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right){\cdot}x\right)}}{\cdot}{2}^{x-1}+\class{steps-node}{\cssId{steps-node-10}{\ln\left(2\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-11}{{2}^{x-1}}}{\cdot}\class{steps-node}{\cssId{steps-node-12}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}{\cdot}{x}^{x}\right)$ $=gt{\cdot}\left(\ln\left(2\right){\cdot}\class{steps-node}{\cssId{steps-node-17}{1}}{x}^{x}{\cdot}{2}^{x-1}+\left(\class{steps-node}{\cssId{steps-node-14}{\class{steps-node}{\cssId{steps-node-13}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}+\class{steps-node}{\cssId{steps-node-16}{x{\cdot}\class{steps-node}{\cssId{steps-node-15}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}\right){\cdot}{x}^{x}{\cdot}{2}^{x-1}\right)$ $=gt{\cdot}\left({x}^{x}{\cdot}{2}^{x-1}{\cdot}\left(\class{steps-node}{\cssId{steps-node-18}{1}}{\cdot}\ln\left(x\right)+\class{steps-node}{\cssId{steps-node-19}{\dfrac{1}{x}}}{\cdot}x\right)+\ln\left(2\right){\cdot}{x}^{x}{\cdot}{2}^{x-1}\right)$ $=gt{\cdot}\left({x}^{x}{\cdot}{2}^{x-1}{\cdot}\left(\ln\left(x\right)+1\right)+\ln\left(2\right){\cdot}{x}^{x}{\cdot}{2}^{x-1}\right)$ Wynik alternatywny: $=gt{x}^{x}{\cdot}{2}^{x-1}{\cdot}\left(\ln\left(x\right)+1\right)+\ln\left(2\right){\cdot}gt{x}^{x}{\cdot}{2}^{x-1}$

$f\left(g, t, x\right) =$

$gt{x}^{x}{\cdot}{2}^{x-1}$

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

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

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

$=gt{\cdot}\left(\class{steps-node}{\cssId{steps-node-5}{\class{steps-node}{\cssId{steps-node-4}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({x}^{x}\right)}}{\cdot}{2}^{x-1}}}+\class{steps-node}{\cssId{steps-node-7}{{x}^{x}{\cdot}\class{steps-node}{\cssId{steps-node-6}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left({2}^{x-1}\right)}}}}\right)$

$=gt{\cdot}\left(\class{steps-node}{\cssId{steps-node-8}{{x}^{x}}}{\cdot}\class{steps-node}{\cssId{steps-node-9}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right){\cdot}x\right)}}{\cdot}{2}^{x-1}+\class{steps-node}{\cssId{steps-node-10}{\ln\left(2\right)}}{\cdot}\class{steps-node}{\cssId{steps-node-11}{{2}^{x-1}}}{\cdot}\class{steps-node}{\cssId{steps-node-12}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x-1\right)}}{\cdot}{x}^{x}\right)$

$=gt{\cdot}\left(\ln\left(2\right){\cdot}\class{steps-node}{\cssId{steps-node-17}{1}}{x}^{x}{\cdot}{2}^{x-1}+\left(\class{steps-node}{\cssId{steps-node-14}{\class{steps-node}{\cssId{steps-node-13}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}{\cdot}\ln\left(x\right)}}+\class{steps-node}{\cssId{steps-node-16}{x{\cdot}\class{steps-node}{\cssId{steps-node-15}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(x\right)\right)}}}}\right){\cdot}{x}^{x}{\cdot}{2}^{x-1}\right)$

$=gt{\cdot}\left({x}^{x}{\cdot}{2}^{x-1}{\cdot}\left(\class{steps-node}{\cssId{steps-node-18}{1}}{\cdot}\ln\left(x\right)+\class{steps-node}{\cssId{steps-node-19}{\dfrac{1}{x}}}{\cdot}x\right)+\ln\left(2\right){\cdot}{x}^{x}{\cdot}{2}^{x-1}\right)$

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

Wynik alternatywny:

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