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.

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,

\(h'(2)=-1\cdot (-1)=1.\)

2. For the function \(g(x)=f(x^2):\)

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

\(g'(x)=(f(x^2))'=f'(x^2)\cdot 2x,\\g'(2)=f'(4)\cdot 4=4f'(4).\)

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

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

facundo3141592 facundo3141592 · Answer 2 · 2021-11-03T05:26:22-04:00

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)(2x)

Then:

g'(2) = f'(2^2)(22)

= f'(4)4 = 1*4 = 4

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