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The function is $\"F(x)=(4x-x^2)^{100}\"$ .

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

Here $\"f(x)=x^{100}\"$ and $\"g(x)=4x-x^2\"$ .

Consider $\"f(x)=x^{100}\"$ .

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

$\"\\small$

Consider $\"g(x)=4x-x^2\"$ .

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

$\"\\small$

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$\"F(x)=(4x-x^2)^{100}\"$ .

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

Substitute $\"\\small$ and $\"\\small$ in chain rule

$\"F\'(x)=100(4x-x^2)^{99}(4-2x)\"$

$\"F\'(x)=(400-200x)(4x-x^2)^{99}\"$ .

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$\"F\'(x)=(400-200x)(4x-x^2)^{99}\"$ .