Proof of the chain rule

In summary: Basically, you take the derivative of y with respect to x, and use that to find the derivative of u with respect to x.
  • #1
zeion
466
1

Homework Statement



I'm looking for a proof for the chain rule that is relatively easy to understand. Can someone show / link me one? Thanks.

Homework Equations





The Attempt at a Solution

 
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  • #2
Google is your friend:

http://math.rice.edu/~cjd/chainrule.pdf
 
  • #3
I posted my calculus notes with a basic (non-rigouous) derivation of the chain rule in, you could have a look at those.
 
  • #4
For the pfd file, I don't understand the middle of page 2 where it says

"..and use the second equation applied to the right-hand-side with k =
[g'(x) + v]h.. Note that using this quantity for k tells us
that k -> 0 as h -> 0, and so w -> 0 as h -> 0."

How did they choose that substitution for k?
 
  • #5
anyone?
 
  • #6
how about this one?
let u(x) be a differentiable fuction in [a,b] with values in [a',b'] and y=f(x) a differentiable fuction in [a',b'].
Let Δx be a randomly picked difference x2-x1. That causes a change Δu on u(x), while Δu causes a change on y=f(u).
We have Δu=(u'(x)+n1)*Δx. You can easily verify by looking at the graph that the line connection the points (x1,u(x1)) and (x2,u(x2)) has a slope equal to the value of the derivative of u on x1 plus a number n1 to compensate for the fact that Δx isn't zero(and thus this line isn't the tanget on x1).
The same applys to Δy=((f'(u)+n2)*Δu.
When Δx-->0 , n1,n2-->0
Δx/Δy=(f;(u)+n2)*(u'(x)+n1)
We calculate the limit of the fraction when Δx-->0 and it is equal to f'(u)*u'(x)=(dy/du)*(du/dx)
system has gone crazy and won't show the math symbols, sorry for the formating

It's the proof from Louis Brand's book "Advanced Calculus", paragraph 52- The chain rule
 

1. What is the chain rule?

The chain rule is a mathematical rule that allows us to find the derivative of a composite function, which is a function made up of two or more other functions. It helps us to calculate the rate of change of one variable with respect to another variable.

2. Why is the chain rule important?

The chain rule is important because it is a fundamental rule in calculus and is used to solve many problems in various fields such as physics, economics, and engineering. It is also essential for finding the derivatives of more complex functions.

3. How do you apply the chain rule?

To apply the chain rule, you need to identify the composite function and break it down into its individual functions. Then, you can use the formula (f ∘ g)'(x) = f'(g(x)) * g'(x) to find the derivative, where f'(x) is the derivative of the outer function and g'(x) is the derivative of the inner function.

4. Can you give an example of using the chain rule?

Sure, let's say we have the function f(x) = (x^2 + 3)^3. Using the chain rule, we can rewrite this as f(x) = u^3, where u = x^2 + 3. Now, we can find the derivatives of both f(x) and u and plug them into the formula (f ∘ g)'(x) = f'(g(x)) * g'(x) to find the derivative of the original function.

5. Are there any shortcuts or tricks for using the chain rule?

Yes, there are some common patterns that can help you to easily identify the inner and outer functions and apply the chain rule. These include the power rule, product rule, quotient rule, and trigonometric identities. Additionally, with practice, you can develop a better understanding of how to apply the chain rule in different situations.

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