f is a differentiable function on the interval [0, 1] and g(x) = f(2x). The table below gives values of f ′(x). What is the value of g ′(0.1)?

x 0.1 0.2 0.3 0.4 0.5
f ′(x) 1 2 3 –4 5
The answers are:
1
2
4
cannot be determined
I got 1 please let me know if this is right or wrong!

g(x)=f(2x)

g'(x)=f'(2x)*2
g'(0.1)=f'(0.2)*2=2*2=4

for example, suppose f(x)=5x^2
Then f'(x) = 10x, as in the table.

Then g(x) = f(2x)=5(2x)^2=20x^2
g'(x) = 40x
g'(.1)=40*.1=4
g'(1) = f'(2)*2 = 2*2=4

Well, let's see if we can help you out with this one! To find the value of g'(0.1), we need to use the chain rule. Since g(x) = f(2x), we can express g(x) as g(x) = f(u), where u = 2x.

To find g'(x), we apply the chain rule, which states that g'(x) = f'(u) * u'. Here, f'(u) is the derivative of f(x) with respect to u, and u' is the derivative of u with respect to x.

In this case, since u = 2x, we have u' = 2. And since f'(x) is given in the table, let's see which value corresponds to u = 2(0.1) = 0.2.

According to the table, f'(0.2) = 2.

Now we can find g'(0.1) by plugging in the values we found into the chain rule equation:

g'(0.1) = f'(u) * u' = f'(0.2) * 2 = 2 * 2 = 4.

So, the value of g'(0.1) is 4.

Therefore, the correct answer is 4. Keep up the good work!

To find the value of g'(0.1), we need to use the chain rule. The chain rule states that if g(x) is a composition of two functions, say g(x) = f(h(x)), then g'(x) = f'(h(x)) * h'(x).

In this case, g(x) = f(2x), which means h(x) = 2x. We know the values of f'(x) at certain points, but we need to find h'(x) in order to find g'(x).

To find h'(x), we differentiate h(x) = 2x with respect to x. The derivative of 2x is 2.

Now, we can use the chain rule: g'(0.1) = f'(h(0.1)) * h'(0.1).

Since h(0.1) = 2 * 0.1 = 0.2, and h'(0.1) = 2, we can substitute these values into the chain rule formula:

g'(0.1) = f'(0.2) * 2.

From the given table, f'(0.2) = 2. Therefore, g'(0.1) = 2 * 2 = 4.

So, the correct answer is 4.

To find the value of g'(0.1), we need to use the Chain Rule in calculus.

The Chain Rule states that if we have a composite function g(x) = f(h(x)), then the derivative of g(x) with respect to x is given by:

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

In this case, we have g(x) = f(2x), so we can think of h(x) = 2x.

To find g'(0.1), we need to find f'(h(x)) and h'(x):

1. Find f'(h(x)):

Since we are given the values of f'(x), we can substitute 2x for x to get f'(2x). The given table provides the values of f'(x) for x = 0.1, 0.2, 0.3, 0.4, and 0.5.

To find the corresponding value of f'(2x) for x = 0.1, we substitute 2 * 0.1 = 0.2 into the table.

We see that the value of f'(2x) for x = 0.1 is 2.

2. Find h'(x):

In this case, h(x) is simply 2x, so h'(x) is the derivative of 2x, which is 2.

Now, we can calculate g'(0.1) using the Chain Rule:

g'(0.1) = f'(h(x)) * h'(x) = f'(2 * 0.1) * 2 = f'(0.2) * 2 = 2 * 2 = 4.

Therefore, the correct answer is 4.

f is a differentiable function on the interval [0, 1] and g(x) = f(4x). The table below gives values of f '(x). What is the value of g '(0.1)?

x 0.1 0.2 0.3 0.4 0.5
f '(x) 1 2 3 –4 5
–16
–4
4
Cannot be determined