Question

Any help with an explanation would be appreciated!

Answer 1

Problem 1

We'll use the product rule to say

h(x) = f(x)*g(x)

h ' (x) = f ' (x)*g(x) + f(x)*g ' (x)

Then plug in x = 2 and use the table to fill in the rest

h ' (x) = f ' (x)*g(x) + f(x)*g ' (x)

h ' (2) = f ' (2)*g(2) + f(2)*g ' (2)

h ' (2) = 2*3 + 2*4

h ' (2) = 6 + 8

h ' (2) = 14

Answer: 14

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Problem 2

Now we'll use the quotient rule

`h(x) = (f(x))/(g(x))\\\\h'(x) = (f'(x)*g(x)-f(x)*g'(x))/((g(x))^2)\\\\h'(2) = (f'(2)*g(2)-f(2)*g'(2))/((g(2))^2)\\\\h'(2) = (2*3-2*4)/((3)^2)\\\\h'(2) = (6-8)/(9)\\\\h'(2) = -(2)/(9)\\\\`

Answer: -2/9

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Problem 3

Use the chain rule

`h(x) = f(g(x))\\\\h'(x) = f'(g(x))*g'(x)\\\\h'(2) = f'(g(2))*g'(2)\\\\h'(2) = f'(3)*g'(2)\\\\h'(2) = 3*4\\\\h'(2) = 12\\\\`

Answer: 12