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

Answer 1

The value of #F'(0)# is #96#.

Consider:

#y = f(g(x))#

Then by the chain rule the derivative is given by #f'(g(x)) * g'(x)#. Thus the derivative of #y = f(g(h(x))# will be derived as being:

#f'(g(h(x)) * "derivative of "g(h(x))#

#f'(g(h(x)) * g'(h(x)) * h'(x)#

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

#F'(x) = f'(3f(4f(x)) * 3f'(4f(x)) * 4f'(x)#

#F'(0) = f'(3f(4f(0)) * 3f'(4f(0)) * 4f'(0)#

Substituting our knowns:

#F'(0) = f'(3f(0)) * 3f'(0) * 4(2)#

#F'(0) = f'(0) * 3(2) * 8#

#F'(0) = 2 * 6 * 8#

#F'(0) = 96#

Hopefully this helps!

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Charles Anctil

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 ]

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

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.