Calculus problem

Question

If h(x) = f[g(x)], use the table of values for f, g, f ' and g ' to find the value of h '(1)

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

1 3 2 2 6

2 1 8 5 7

3 7 2 7 9

Answer 1

Given h(x) = f [g(x)] , find h'(x)

Since h(x) is a composite function (a function that is made of 2 or more functions with one of those functions inside of the other), you must use the Chain Rule. A good way to remember the Chain Rule is:

D_Outer·D_Inner

_{Take the derivative of the outer function, leaving the inner function unchanged. Then multiply by the derivative of the inner function.}

_{h'(x) = f'[g(x)]·g'(x)}

_{Now to find h'(1), we need a few things...}

_{h'(1) = f'[g(1)]·g'(1)}

Work from the inside out and using the chart from above to find the function values at x = 1:

g(1) = 2

g'(1) = 6

Now note that f' needs g(1) as its input and not x = 1. Well we found g(1) = 2. So...

f'[g(1)] = f'(2) = 5

And...

h'(1) = f'[g(1)]·g'(1) = 5(6) = 30

Answer 2

Since h(x) = f(g(x)), then we would need to use the chain rule to find the derivative of h(x) in terms of f and g.

The derivative of f(g(x)) = f '(g(x))*g'(x) so h'(1) = [f(g(1))] ' = f '(g(1))*g'(1) = f '(2)*6 = 5*6 = 30