# If #F(x)=f(3f(4f(x)))# where f(0)=0 and f'(0)=2, how do you find F'(0)?

The value of

Consider:

If we return our attention to the problem at hand, we get:

Substituting our knowns:

Hopefully this helps!

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To find ( F'(0) ), apply the chain rule and the given conditions for ( f(0) ) and ( f'(0) ):

[ F'(x) = f'(3f(4f(x))) \cdot 3f'(4f(x)) \cdot 4f'(x) ]

Substitute ( x = 0 ) and the given conditions for ( f(0) ) and ( f'(0) ):

[ F'(0) = f'(3f(4f(0))) \cdot 3f'(4f(0)) \cdot 4f'(0) = f'(0) \cdot 3f'(0) \cdot 4f'(0) = 2 \cdot 3 \cdot 4 = 24 ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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