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The function is $\"f(z)=\\frac{1}{z^2+1}\"$ .

The function is in the form of composite function $\"h\\$ .

Here $\"g(z)=z^2\"$ and $\"h(z)=\\frac{1}{(1+z)}\"$ .

Consider $\"g(z)=z^2\"$ .

Differentiate on each side with respect to $\"z\"$ .

$\"\\small$

$\"\\small$ .

Consider $\"h(z)=\\frac{1}{(1+z)}\"$ .

Differentiate on each side with respect to $\"z\"$ .

$\"\\\\h\'(z)=(-1)(1+z)^{-2}\\\\$

$\"\"$

$\"f(z)=\\frac{1}{z^2+1}\"$ .

Chain Rule of derivatives: $\"\\frac{d}{dx}h(g(z))=$ .

Substitute $\"\\\\h\'(z)=\\frac{-1}{(1+z)^{2}}\"$ and $\"\\small$ in chain rule.

$\"\\\\f\'(z)=\\frac{-1}{(1+z^2)^{2}}(2z)\\\\$

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$\"\\\\f\'(z)=\\frac{-2z}{(1+z^2)^{2}}.\"$