Finding F'(5) from F(x) = f(g(x))

Thread starter TsAmE
Start date Apr 19, 2010

In summary, the chain rule is a method for finding the derivative of a composite function, where one function is nested inside another. It is important to use the chain rule when finding F'(5) because it simplifies the process and saves time. However, it is important to avoid common mistakes such as forgetting to take the derivative of the inner function and using the wrong value for g(5). The chain rule can be used for any type of composite function, as long as the functions are differentiable.

Apr 19, 2010

TsAmE

Homework Statement

If F(x) = f(g(x)), where f(-2) = 8, f'(-2) = 4, f'(5) = 3, g(5) = -2, and g'(5) = 6, find F'(5).

Homework Equations

F'(x) = f'(g(x)) * g'(x) - Chain Rule

The Attempt at a Solution

F'(x) = f'(g(x)) * g'(x)
F'(5) = f'(g(5)) * g'(5)
F'(5) = f'(-2) * 6
F'(5) = 8 * 6
F'(5) = 48

The correct answer is 24

which doesn't make sense.

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Apr 19, 2010

lanedance

Homework Helper

f'(-2) = 4

1. What is the process for finding F'(5) from F(x) = f(g(x))?

To find F'(5), you will need to follow the chain rule, which states that the derivative of F(x) = f(g(x)) is equal to f'(g(x)) * g'(x). First, find the derivative of f(x) and g(x) separately. Then, plug in the value of g(5) into f'(x) and g'(x) and multiply them together. The resulting value will be F'(5).

2. Can you explain the chain rule in simpler terms?

The chain rule is a method for finding the derivative of a composite function, where one function is nested inside another. It tells us that the derivative of the outer function is equal to the derivative of the inner function multiplied by the derivative of the outer function with the inner function plugged in.

3. Why is it important to use the chain rule when finding F'(5)?

The chain rule is important because it allows us to find the derivative of a composite function without having to go through the process of finding the derivative of each individual function and then combining them. It saves time and simplifies the process.

4. Are there any common mistakes to avoid when using the chain rule to find F'(5)?

One common mistake to avoid is forgetting to take the derivative of the inner function. It is important to remember to find the derivative of both the inner and outer functions separately. Another mistake is plugging in the wrong value for g(5) into f'(x) and g'(x). Make sure to use the correct value for g(5) to avoid incorrect results.

5. Can the chain rule be used for any type of composite function?

Yes, the chain rule can be used for any type of composite function, as long as the functions are differentiable. This means that the functions must be continuous and have a defined derivative at every point in the domain.