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Pochodna funkcji arctg(x)-arctg(1/x)

$f\left(a, c, g, r, t, x\right) =$	$acgrtx-\dfrac{acgrt}{x}$
$\dfrac{\mathrm{d}\left(f\left(a, c, g, r, t, x\right)\right)}{\mathrm{d}x} =$	$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(acgrtx-\dfrac{acgrt}{x}\right)}}$ $=\class{steps-node}{\cssId{steps-node-2}{acgrt-acgrt{\cdot}\class{steps-node}{\cssId{steps-node-3}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(\dfrac{1}{x}\right)}}}}$ $=acgrt{\cdot}\left(1-\dfrac{\class{steps-node}{\cssId{steps-node-6}{-\class{steps-node}{\cssId{steps-node-5}{\tfrac{\mathrm{d}}{\mathrm{d}x}\kern-.25em\left(x\right)}}}}}{\class{steps-node}{\cssId{steps-node-4}{{x}^{2}}}}\right)$ $=acgrt{\cdot}\left(\dfrac{\class{steps-node}{\cssId{steps-node-7}{1}}}{{x}^{2}}+1\right)$ Wynik alternatywny: $=\dfrac{acgrt}{{x}^{2}}+acgrt$

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

$acgrtx-\dfrac{acgrt}{x}$

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

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

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

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

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

Wynik alternatywny:

$=\dfrac{acgrt}{{x}^{2}}+acgrt$