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Pochodna funkcji (sin(x))^tan(x)

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

$f\left(x\right) =$

${\left(\sin\left(x\right)\right)}^{\tan\left(x\right)}$

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

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

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

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

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

$={\left(\sin\left(x\right)\right)}^{\tan\left(x\right)}{\cdot}\left({\left(\sec\left(x\right)\right)}^{2}{\cdot}\ln\left(\sin\left(x\right)\right)+\dfrac{\class{steps-node}{\cssId{steps-node-11}{\cos\left(x\right)}}{\cdot}\tan\left(x\right)}{\sin\left(x\right)}\right)$