Using the mean value theorem to prove the chain rule

In summary, the conversation discusses the use of the mean value theorem to prove the chain rule for functions in C1. Specifically, it is mentioned that g and f are functions taking open subsets I and J of the real line respectively, with g mapping J to R. The conversation also mentions the use of the mean value theorem and the equation (g o f)' (x) = g'(f (x)) f'(x) in the attempt at a solution.
  • #1
B3NR4Y
Gold Member
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Homework Statement


I and J are open subsets of the real line. The function f takes I to J, and the function g take J to R. The functions are in C1. Use the mean value theorem to prove the chain rule.

Homework Equations


(g o f)' (x) = g'(f (x)) f'(x)
MVT

The Attempt at a Solution


[/B]
I know that the open interval (an, x) is a subset of J.

Therefore I can apply the mean value theorem, but I have no clue where to go
 
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  • #2
B3NR4Y said:

Homework Equations


(g o f)' (x) = g'(f (x)) f'(x)
This is slightly misleading. I prefer [itex]\frac{dg}{dx}=\frac{dg}{df}\cdot \frac{df}{dx} [/itex]. That should give you a clue.
 

FAQ: Using the mean value theorem to prove the chain rule

1. What is the mean value theorem?

The mean value theorem is a fundamental theorem in calculus that states that for a differentiable function on a closed interval, there exists a point in that interval where the slope of the tangent line is equal to the average rate of change of the function over that interval.

2. How is the mean value theorem related to the chain rule?

The chain rule is a rule in calculus that allows us to find the derivative of a composite function. The mean value theorem is used to prove the chain rule by showing that the derivative of a composite function can be expressed as the product of the derivative of the outer function and the derivative of the inner function evaluated at a specific point.

3. Can you provide an example of using the mean value theorem to prove the chain rule?

Yes, for example, we can use the mean value theorem to prove the chain rule for the function f(x) = (x^2 + 1)^3. First, we let g(x) = x^2 + 1 and h(x) = x^3. Then, applying the mean value theorem to g(x) on the interval [a, b], we get g'(c) = (g(b) - g(a))/(b - a) for some c in [a, b]. Substituting this into the chain rule, we get f'(x) = g'(c) * h'(x) = (3c^2)(3x^2) = 9c^2x^2, which is the same as the derivative of f(x).

4. Can the mean value theorem be used to prove other calculus rules?

Yes, the mean value theorem can be used to prove other important calculus rules, such as Rolle's theorem and the first and second derivative tests for extrema.

5. What is the significance of using the mean value theorem to prove the chain rule?

The mean value theorem provides a rigorous proof for the chain rule, which is a crucial tool in calculus for finding the derivatives of composite functions. It also helps us to better understand the relationship between the instantaneous rate of change and the average rate of change of a function.

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