Question

the table above gives values of the differentiable functions f and g and of their derivatives f′ and g′ , at selected values of x. if h(x) = f(g(x)), what is the slope of the graph of h at x = 2?

Answer 1

Explanation:

Chain Rule:

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

We'll start with the outside function first f(g(x)), and take the derivative to get f'(g(x))

Then we'll take the inside function of g(x), take the derivative and get g'(x)

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

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

Using the chart, we can plug in the values and simplify.

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

h(2) = f'(-1) * (2)

h(2) = 3 * 2

h(2) = 6

Answer 2

The slope of graph of the h at is 6.

Consider the given table of values

Given h(x) = f(g(x))

Need to find the slope of graph of h at x = 2

We know that d dx ( f(g(x)) = f' * (g(x)) * g' * (x)

Now h(x) = f(g(x))

Rightarrow h' * (x) = f' * (g(x)) * g' * (x) -> (1)

From table we have g(2) = - 1, f' * (- 1) = 3 and

Substitute x = 2 in the equation (1)

Then, h' * (2) = f' * (g(2)) * g' * (2)

= f' * (- 1) * 2

= 3× 2

=6

Hence, The slope of graph of the h at x=2 is h(2) = 6.