Suppose the functions f and g and their derivatives have the following values at x = 1 and x = 2. Let h(x) = f(g(x)). Evaluate h′(1).
X | f(x) G(x) f'(x) g'(x)
____________________________
1 | 8 2 1/3 -3
2 | 3 -4 2π 5
My answer: -6π
Suppose the functions f and g and their derivatives have the following values at x = 1 and x = 2. Let h(x) = f(g(x)). Evaluate h′(1).
X | f(x) G(x) f'(x) g'(x)
____________________________
1 | 8 2 1/3 -3
2 | 3 -4 2π 5
f
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To find h′(1), we need to use the chain rule.
The chain rule states that if h(x) = f(g(x)), then h′(x) = f′(g(x)) * g′(x).
Given the values at x = 1, we have:
f(1) = 8
g(1) = 2
f′(1) = 1/3
g′(1) = -3
Using the chain rule:
h′(1) = f′(g(1)) * g′(1) = f′(2) * (-3)
Looking at the table, we see that f′(2) is not given. However, we can calculate it using the values at x = 2:
f(2) = 3
g(2) = -4
f′(2) = 2π
g′(2) = 5
Therefore, h′(1) = f′(2) * (-3) = (2π) * (-3) = -6π.
So, h′(1) = -6π.
To evaluate h'(1), we need to find the derivative of h(x) with respect to x and then evaluate it at x = 1.
We know that h(x) = f(g(x)), so we need to find f'(g(x)) and g'(x) in order to apply the chain rule.
Let's first find f'(g(x)). We are given that f'(x) = 1/3, and g(x) = 2 when x = 1. So, substituting g(x) = 2 in f'(x), we have:
f'(g(x)) = 1/3.
Next, we need to find g'(x) in order to apply the chain rule. We are given that g'(x) = 5 when x = 1.
Now, we can evaluate h'(x) by applying the chain rule:
h'(x) = f'(g(x)) * g'(x).
Plugging in the values, we have:
h'(x) = (1/3) * 5 = 5/3.
Finally, we evaluate h'(x) at x = 1:
h'(1) = 5/3.
Therefore, the value of h'(1) is 5/3.