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Pochodna funkcji ln(tan(1/4pi+1/2x))

$f\left(x\right) =$	$\ln\left(\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)$ Note: Your input has been rewritten/simplified.
$\dfrac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{\dfrac{1}{\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)}}}{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)}}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-4}{{\left(\sec\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)}^{2}}}{\cdot}\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)}}}{\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)}$ $=\dfrac{\class{steps-node}{\cssId{steps-node-6}{\dfrac{1}{2}}}{\cdot}{\left(\sec\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)}^{2}}{\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)}$ $=\dfrac{{\left(\sec\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)}^{2}}{2{\cdot}\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)}$

$f\left(x\right) =$

$\ln\left(\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)$

Note: Your input has been rewritten/simplified.

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

$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\ln\left(\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)\right)}}$

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

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

$=\dfrac{\class{steps-node}{\cssId{steps-node-6}{\dfrac{1}{2}}}{\cdot}{\left(\sec\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)}^{2}}{\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)}$

$=\dfrac{{\left(\sec\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)\right)}^{2}}{2{\cdot}\tan\left(\dfrac{x}{2}+\dfrac{{\pi}}{4}\right)}$