Let `F(x)=f(f(x))` and `G(x)=F(x)^2` . Knowing `f(4)=5, f(5)=3, f'(5)=8` and `f'(4)=5` , find `F'(4)` and `G'(4).`

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We need the chain rule for both. According to that, we get

`F'(4)=f'(f(4))*f'(4)=f'(5)*f'(4)=8*5=40` and

`G'(4)=2F(4)*F'(4)=2*f(f(4))*40=2*f(5)*40=6*40=240.`

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