If f is the function whose graph is shown below, let h(x) = f(f(x)) and g(x) = f(x2). use the graph of f to estimate the value of each derivative.

We want to estimate the value of two derivates by knowing the graph of the function f(x) and the derivation rules.

The estimates are:

g'(2) = 4

h'(2) = 0.5

We know that the derivation of a composition:

k(x)  = f(g(x))

is given by:

k'(x) = f'(g(x))*g'(x).

Now we have:

f(x) is the function shown in the graph.

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

g(x) = f(x^2)

Now we want to get both of these derivated and evaluated in x = 2.

We will get:

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

Evaluating that in x = 2 we get:

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

By looking at the graph we estimate:

f(2) = 1

f'(2) = -1

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

The derivate at x = 1 is something near -0.5, then we got:

h'(2) = -0.5*-1 = 0.5

For g(x) we have:

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

Then:

g'(2) = f'(2^2)*(2*2)

       = f'(4)*4 = 1*4 = 4

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1. For the function h(x)=f(f(x)):

Use the chain rule to determine the derivative - h'(x):

From the graph f(2)=1, f'(f(2))=f'(1)=-1 (the tangent line to f(x) at x=1 has a slope -1) and f'(2)=-1 (the tangent line to f(x) at x=2 has a slope -1). Therefore,

2. For the function

Use the chain rule to determine the derivative - g'(x):

From the graph f'(4)=1 (the tangent line to f(x) at x=4 has a slope 1) and g'(2)=4.

h'(2)=1, g'(2)=4.