If F(x) = f (g (x) ), where f(-2) = 8, f'(-2) = 4, f'(5) = 3, g(5) = -2, g'(5) = 6. Then F'(5) is ??

Answer 1

I'm going to lay this out in a chart so it's a little easier to see:

F(x) = f(g(x))

x | f (x) | f ' (x) | g (x) | g ' (x)

--------------------------------------

-2 | 8 | 4 |

5 | | 3 | -2 | 6

Remember the chain rule, which says

(f (g (x))) ' = g ' (x) f ' (g (x))

When they ask for F ' (5), they are asking for (f (g (x))) ' when x = 5.

Using the chain rule, that's

F ' (5) = g ' (5) f ' (g (5))

We can simplify using the numbers provided.

F ' (5) = (6) f ' (-2)

F ' (5) = (6) (4)

F ' (5) = 24

I hope that helps!

~ Logan

Answer 2

If F(x) = f (g (x) ), then by the Chain Rule

F ' (x) = f ' (g(x)) g ' (x)

Now is simply a matter of plugging in numbers.

F ' (5) = f ' (g(5)) g ' (5)

When x = 5, g(5) = -2 and g ' (5) = 6. So

F ' (5) = 6 f ' (-2) = 6 • 4 = 24

Answer 3

F(x) = f(g(x))

Using chain rule:

u = g(x)

F(x) = f(u)

F'(x) = f'(u) * g'(x)

u = g(5) = -2

g'(5) = 6

f'(u) = f'(-2) = 4

F'(5) = f'(u) * g'(5) = 4 * 6 = 24